A TTI A CCADEMIA N AZIONALE L INCEI C LASSE S CIENZE F ISICHE M ATEMATICHE N ATURALI



        R ENDICONTI L INCEI
    M ATEMATICA E A PPLICAZIONI



     Enrico Bombieri

     Remarks on Weil’s quadratic functional in the
     theory of prime numbers, I

     Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche,
     Matematiche e Naturali. Rendiconti Lincei. Matematica e
     Applicazioni, Serie 9, Vol. 11 (2000), n.3, p. 183–233.
     Accademia Nazionale dei Lincei
     <http://www.bdim.eu/item?id=RLIN_2000_9_11_3_183_0>




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Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e
Naturali. Rendiconti Lincei. Matematica e Applicazioni, Accademia Nazionale dei
Lincei, 2000.
Rend. Mat. Acc. Lincei
s. 9, v. 11:183-233 (2000)



                  Remarks on Weil’s quadratic functional
                    in the theory of prime numbers, I

                                   Memoria (*) di Enrico Bombieri


     Abstract. — This Memoir studies Weil’s well-known Explicit Formula in the theory of prime numbers
and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis
is true. We prove that this quadratic functional attains its minimum in the unit ball of the L2 -space of
functions with support in a given interval [−t; t ], and prove again Yoshida’s theorem that it is positive
definite if t is sufficiently small. The Fourier transform of the functional gives rise to a quadratic form
in infinitely many variables and we then study its finite truncations and corresponding eigenvalues. In
particular, if the Riemann Hypothesis is false but only with finitely many non-trivial zeros off the critical
line we show that the number of negative eigenvalues is precisely one-half of the number of zeros failing to
satisfy the Riemann Hypothesis, provided the truncation is big enough.


    Key words: Prime number theory; Riemann Hypothesis; Explicit Formula.


     Riassunto. — Osservazioni sul funzionale quadratico di Weil nella teoria dei numeri primi, I. Questa
Memoria studia la nota Formula Esplicita di Weil nella teoria dei numeri primi e il funzionale quadratico
associato ad essa. Questo funzionale è positivo semidefinito se e solo se l’Ipotesi di Riemann è valida.
Dimostriamo qui che il minimo di questo funzionale nello spazio delle funzioni L2 con supporto compatto
nell’intervallo [−t; t ] è raggiunto, e dimostriamo nuovamente il risultato di Yoshida che dà la positività
per t sufficientemente piccolo. La trasformata di Fourier del funzionale dà luogo ad una forma quadratica
in un numero infinito di variabili, e ne studiamo i suoi troncamenti finiti e gli autovalori corrispondenti.
In particolare, se l’Ipotesi di Riemann è falsa ma solamente con un numero finito di eccezioni, si dimostra
che il numero di autovalori negativi è la metà del numero di eccezioni all’Ipotesi di Riemann, purché il
troncamento sia abbastanza grande.


                                            1. Introduction
   The Explicit Formula in the theory of prime numbers is a generalization of Rie-
mann’s famous exact formula of 1859 expressing the number of primes up to a given
limit in terms of a sum of a certain Mellin transform, evaluated at the zeros of the
Riemann zeta function. Since then, it has found wide use in analytic number the-
ory. However, most applications found in the literature involve the use of well-chosen
explicit test functions and use only approximations, not exact evaluations.
   In 1942, A.P. Guinand [2] studied for the first time the Explicit Formula in a
general setting, viewing it as a transformation formula not unlike Poisson Summation
Formula.
   In 1952, A. Weil [6] (1) put forward a more general Explicit Formula which had

   (*) Pervenuta in forma definitiva all’Accademia il 7 settembre 2000.
   (1) It appears that Weil was either unaware of Guinand’s work or dismissed it as uninteresting. Certainly
Weil’s aim in his paper was entirely different from Guinand’s.
184                                                                              e. bombieri

an identical formulation both in the classical case and the so-called function field case.
The main point of Weil’s paper was to highlight the deep analogies between classical
zeta functions and congruence zeta functions arising from function fields of curves over
a finite field of positive characteristic.
     Since Weil had already proved the analogue of the Riemann Hypothesis in the
function field case, it was natural to ask if the classical Riemann Hypothesis could also
be interpreted in the same light. Thus in the same paper Weil formulated the Riemann
Hypothesis as the positivity of a certain quadratic functional arising from the Explicit
Formula.
     Even if Weil’s work has since then been vastly generalized and reinterpreted in the
framework of adeles, nothing of consequence in prime number theory has emerged
so far from a direct study of Weil’s functional, which unfortunately appears to be as
intractable as the Riemann Hypothesis itself. A first study of Weil’s functional was
done by H. Yoshida [5] in 1992. In his paper, Yoshida studies the behaviour of the
associated hermitian form in certain Hilbert spaces of functions supported in an interval
[−t; t ], obtaining several interesting results. Besides reproving Weil’s criterion for the
validity of the Riemann hypothesis, he shows that the positivity of the functional in
the class of smooth even functions with compact support is equivalent to the validity
of the Riemann hypothesis excluding real zeros. Moreover, he shows how the positivity
of this functional for functions supported in a fixed interval [−t; t ] can be reduced to
a finite calculation (depending on t ), and verifies this positivity for t = (log 2)=2.
     In this paper, we study the Weil functional from a variational point of view, restricted
to two classes of Hilbert spaces each of which is sufficiently wide for testing the validity
of the Riemann Hypothesis. We verify that the infimum of the Weil functional in the
unit sphere of spaces in the second class is always attained, and also in spaces of the
first class if the infimum in question is negative. It is noteworthy that the proof of the
first result hinges on the special structure of the term for the «prime at infinity» in the
Explicit Formula.
     Next, we examine the Fourier transform of the functional, which is easily interpreted
as a quadratic form in infinitely many variables. This leads to the study of the eigen-
values of certain infinite matrices, as well as of their finite dimensional truncations. In
view of the special structure of these matrices, it turns out that the eigenvalue 0 cannot
occur for the finite dimensional approximations. From this result, which appears to
be new, we deduce the invariance of the number of negative eigenvalues under cer-
tain deformations of the matrices, thereby determining the exact number of negative
eigenvalues.
     The results obtained for the finite dimensional case carry over to the infinite di-
mensional case, provided we deal with negative eigenvalues bounded away from 0 and
provided their number remains bounded. For the applications we have in mind, the lat-
ter condition follows from the assumption that there are only finitely many non-trivial
zeros of ζ (s) not on the critical line. The existence of a negative eigenvalue follows
again from our stability result in the finite dimensional case, and the main question
here is to decide whether or not this negative eigenvalue stays bounded away from zero,
remarks on weil’s quadratic functional in the theory of prime numbers, I                       185


as we approximate the infinite dimensional matrix by its finite dimensional truncations.
   The upshot is that if the negative eigenvalue stays bounded away from zero then the
Weil functional, restricted to the Hilbert space determining the infinite matrix, is not
positive definite there, while on the other hand if the negative eigenvalue tends to 0
then we obtain in the limit a non-trivial linear relation among monomials x −ρ , where
ρ ranges over the non-trivial zeros of the Riemann zeta function.
   Rather than giving in this introduction a summary of every result, the following
corollary of one of our theorems best illustrates what is obtained here in the end. We
show that one of the following statements holds:
   (i) the Riemann Hypothesis is true ;
   (ii) there are infinitely many complex zeros ρ of the Riemann zeta function with (ρ) = 12 ;
   (iii) there is a linear combination
                                       cρ
                                              x −ρ + A + B x −1
                                     ρ(1 − ρ)
                                 ρ
     
with |cρ |2 = 1 and vanishing identically for 1 ≤ x ≤ M0 , where M0 > 1 is an explicitly
computable constant. Moreover, at least 1=2 of the ‘2 -mass of the coefficients is supported on the
non-trivial zeros of ζ (s) off the critical line.
     More precise results of this type are contained in §10, Theorem 10 and §11, The-
orem 11.
     The question of linear independence which arises in (iii) is of some interest and,
although probably quite difficult, deserves study. Linear relations such as in (iii) above
do occur, but what we obtain here are well-determined relations arising from a specific
limiting process, and they may carry interesting information about consequences of a
hypothetical failure of the Riemann Hypothesis.
     The content of this paper is as follows. In Sections 2 and 3 we restate the Guinand-
Weil Explicit Formula and Weil’s formulation of the Riemann Hypothesis as the posi-
tivity of a certain quadratic functional. Section 4 introduces two variational eigenvalue
problems associated to Weil’s quadratic functional and proves the existence of extremals
for the second problem. Sections 5 and 6 study the structure of extremals and a re-
formulation of the eigenvalue problems, obtained by taking Fourier transforms. Section
7 shows that the first eigenvalue problem admits a resolvent and determines some of
its properties, although results in this section are not used anywhere else in this paper.
Section 8 introduces finite approximations, odd and even eigenfunctions and proves a
key result (Theorem 8) about eigenvalues of finite matrices in the approximation of the
first eigenvalue problem. The main result of this section is quite general and does not
depend on properties of zeta functions or arithmetic. Section 9 obtains corresponding
results for the second eigenvalue problem. Section 10 shows how to pass to the limit
from the finite approximations to the case of interest, namely zeta functions. However,
the possibility that in the limiting process a negative eigenvalue may have limit 0 gives
rise to linear dependence relations, as in alternative (iii) above. Section 11 carries the
same limiting process for the second eigenvalue problem and gives an example of how
186                                                                                e. bombieri

linear relations may arise in the case of zeta functions. Finally, Section 12 proves again
Yoshida’s result of the positivity of Weil’s quadratic functional for test functions with
sufficiently small compact support.
    In this paper we have not attempted to achieve maximum generality or the sharpest
possible statements. All results of this paper extend, mutatis mutandis, to Dedekind zeta
functions of number fields. It should also be possible to replace the assumption that
there are only finitely many non-trivial zeros of ζ (s) off the critical line, which is used
in several places, by a suitable density hypothesis.

                                2. The Guinand-Weil Explicit Formula
   We state the Explicit Formula, in the special case of the Riemann zeta function, in
the following form.
   For a function f (x) on (0; ∞) we define f ∗ by the formula
                                              
                                  ∗       1     1
                                 f (x) = f          ;
                                          x     x
and say that f is even if f = f ∗ and odd if f = −f ∗ .
   Explicit Formula. Let f (x) ∈ C0∞ ((0; ∞)) be a smooth complex-valued function with
compact support in (0; ∞).
   Let                                                 ∞
                                         f(s) =            f (x) x s −1 dx
                                                    0

be the Mellin transform of f . Then we have (2)
                      ∞             ∞              
                                                      ∞

             f(ρ) =      f (x) dx +     f ∗ (x) dx −   Λ(n) {f (n) + f ∗ (n)} −
             ρ              0                 0                     n =1
                                                           ∞                       
                                                                               2        x dx
                                   − (log 4π + γ )f (1) −     f (x) + f ∗ (x) − f (1) 2
                                                           1                   x       x −1
where the first sum ranges over all complex zeros of the Riemann zeta function.
   Moreover, the last two terms in the right-hand side of the Explicit Formula can be
written as                               
                                      1           Γ w
                    −(log π )f (1) +                    f(w) dw:
                                     2π i ( 1 )   Γ 2
                                                        2

   Proof. The proof is an application of the calculus of residues. An elementary treat-
ment is as follows.
   We write as usual σ = (s) and t = (s) for the real and imaginary part of the
complex variable s. Let
                                          Z (s) = π −s=2 Γ(s=2)ζ (s)

      (2) Here γ is Euler’s constant γ = 0:5772156649 : : : .
remarks on weil’s quadratic functional in the theory of prime numbers, I                             187


and consider the integral
                                                  
                                            1         Z
(2.1)                             I (f ) =               (w) f(w) dw
                                           2π i   (c) Z

where the integration is over the line (c − i ∞; c + i ∞) and c > 1. Since f (x) is smooth
with compact support, its Mellin transform f(s) is an entire function of s of order 1
and exponential type, rapidly decreasing in every fixed vertical strip.
   The logarithmic derivative of Z (s) is holomorphic for σ > 1 and has logarithmic
growth on any vertical line (c − i ∞; c + i ∞) with c > 1, hence the above integral is
absolutely convergent.
   For (w) > 1 we have
                    Z           1             1 Γ w      ∞
                                                                Λ(n)
                        (w) = − (log π ) +               −           ;
                     Z           2             2 Γ 2             nw
                                                                    n =1

whence
                                                              
                      1           1                     1          Γ w 
         I (f ) = −     (log π )            f(w) dw +                    f (w) dw −
                      2          2π i   (c)            4 πi    (c) Γ   2
                                                                    ∞          
                                                                             1
                                                                 −     Λ(n)          f(w) n−w dw;
                                                                            2π i (c)
                                                                   n =1

because term-by-term integration is justified by total convergence. The inverse Mellin
transform formula is                      
                                      1
                            f (x) =           f(w) x −w dw
                                     2π i (c)
and the formula for I (f ) becomes
                                                                               
                                                                                ∞
                       1                  1               Γ   w
(2.2)        I (f ) = − (log π ) f (1) +                           f(w) dw −          Λ(n) f (n):
                       2                 4π i         (c) Γ    2
                                                                                n =1

    Now we compute I (f ) in another way. In (2.1), we move the line of integration
to the left, to a line (c  − i ∞; c  + i ∞) with c  < 0. This requires some justification,
namely integrating over a rectangle with vertices at (−c  − iT; c − iT; c + iT; c  + iT )
and showing that the integral over the horizontal sides tends to 0 if we let T go to
infinity along a well-chosen sequence {Tν }. For this step, which is very classical, we
refer to Ingham’s excellent Cambridge Tract [3].
    In moving the line of integration to the left we encounter the residues of Z  =Z (w)
due to the simple poles of Z (w) at w = 0 and w = 1 and to the zeros of Z (w) inside
the critical strip 0 ≤ (w) ≤ 1. It follows that
                                                          
                                                     1          Z
(2.3)          I (f ) = −f (0) − f (1) +      f (ρ ) +                (w) f(w) dw:
                                                       2π i (c  ) Z
                                            ρ

   Now we use the functional equation Z (w) = Z (1 − w), which for (w) < 0 gives us
                                                              ∞
        Z          Z            1          1 Γ 1 − w              Λ(n)
           (w) = − (1 − w) = (log π ) −                      +            :
         Z          Z             2          2 Γ       2
                                                                n =1
                                                                     n1−w
188                                                                                                           e. bombieri

We substitute into (2.3) and obtain
                                    
          I (f ) = −f(0) − f(1) +   f(ρ) +
                                               ρ


                                                                                                  
                       1           1                             1                  Γ       1−w
                   +     (log π )                    f(w) dw −                                          f(w) dw +
                       2          2π i        (c  )            4π i         (c  ) Γ         2

                       
                       ∞                  
                                    1
                   +          Λ(n)                 f(w) n−1+w dw =
                                   2π i   (c  )
                       n =1

                                          
                = −f(0) − f(1) +                 f(ρ) +
                                               ρ


                                                                                                   
                                                                                                      ∞
                    1                  1                          Γ       1−w
                   + (log π ) f (1) −                                                f(w) dw +              Λ(n) f ∗ (n);
                    2                 4π i                 (c  ) Γ         2
                                                                                                      n =1



again because term-by-term integration is justified by total convergence.
   We compare this formula with formula (2.2) for I (f ) and find
                                                  
                                                   ∞

                f(ρ) = f(0) + f(1) −                   Λ(n) {f (n) + f ∗ (n)} − (log π ) f (1) +
            ρ                                      n =1
(2.4)
                                                                                                          
                                1         Γ       w                   1                    Γ       1−w
                         +                                 f(w) dw +                                            f(w) dw:
                               4π i   (c) Γ        2                  4π i           (c  ) Γ         2

     In order to obtain the explicit formula we compute the last two integrals as follows.
First of all, we move the line of integration in the last two integrals to c = 12 and
c  = 12 , which we may without encountering any pole of the integrand. Thus the sum
of the two integrals becomes
                           ∞                             
                        1           Γ 1     i        1
                                         + v f         + iv dv:
                       2π −∞        Γ 4      2        2
This already proves the Explicit Formula with the last two terms expressed in the
alternative way by means of a complex integral.
    Now we recall that [7, Ch. XII, §12.16, p. 241]
                                            ∞           
                        Γ            1  1           1
                           (z) = −γ − +            −
                        Γ             z          n   n+z
                                                              n =1

and
                                                      
                          1        1                  1
                       1 + + ::: +   = log N + γ + O     ;
                          2        N                  N
remarks on weil’s quadratic functional in the theory of prime numbers, I                     189


thus
                                                 N
                                                  1                
                                Γ                          1 + |z |
                                   (z) = log N −       +O
                                Γ                  n+z        N
                                                   n =0

uniformly for (z) ≥ −N=2, z not a negative integer or 0. This gives
                                    N                                  
             Γ 1     i                        4n + 1             1 + |v |
                   + v     = log N −                        +O
              Γ 4     2
                                      n =0
                                           (2n + 12 )2 + v 2        N

and
              ∞                                     
         1             Γ       1  i              1
                                 + v        f     + iv dv =
        2π   −∞        Γ        4  2              2

                             ∞                  N
                                                                                 
                      1                             4n + 1                  1
                   =                  log N −                         f     + iv dv +
                     2π                                1 2                   2
                             −∞               n=0 2n + 2   + v2

                                                                ∞                         
                                                                    1 + |v |   1        
                                                          +O                   f     + iv  dv :
                                                                 −∞   N            2

   Since f is rapidly decreasing on any vertical line, the last integral converges and the
O( ) term is indeed O(1=N ). Also
                                  ∞
                               1         1
                                      f( + iv) dv = f (1);
                              2π −∞ 2
whence
               ∞                                        
          1                 Γ       1  i             1
                                      + v       f     + iv dv =
         2π     −∞          Γ        4  2             2
(2.5)
                                       N
                                           ∞                                     
                                         1            4n + 1          1 + iv dv + O 1 :
               = (log N ) f (1) −                                   f
                                            2π −∞ (2n + 12 )2 + v 2     2            N
                                       n =0

    We have by Fubini’s theorem
           ∞                                ∞        ∞
        1        2a  1                    1        2a
                       f     + iv dv =                      f (x)x −1=2+iv dx dv =
       2π −∞ a 2 + v 2     2             2π −∞ a 2 + v 2 0
(2.6)
                                 ∞                 ∞                  
                                           −1=2 1         1         1
                              =     f (x)x                     −            x iv dv dx:
                                 0             2π i −∞ v − ia v + ia
An easy application of the calculus of residues [7, Ch. VI, §6.22, pp. 113-114] shows
that for (a) > 0 we have
                       ∞                    
                   1            1         1
                                    −           x iv dv = min(x; 1=x)a
                  2π i −∞ v − ia v + ia
190                                                                              e. bombieri

hence by (2.5) and (2.6) we find
          ∞                                  
       1         Γ 1      i            1
                        + v f              + iv dv =
      2π −∞       Γ 4      2             2
                      ∞     N
                                                        1   N
                                                                        
                                    −2n          dx                  2n
                 =−               x       f (x)      −             x      f (x) dx +
                       1                          x      0
                             n =0                             n =0
(2.7)                                      
                                              1
                   + (log N ) f (1) + O             =
                                              N
                      ∞     N
                                                                                      
                                    −2n
                                                     ∗
                                                            dx                        1
                 =−               x         f (x) + f (x)        + (log N ) f (1) + O     :
                       1                                     x                         N
                               n =0

Finally, we have
 ∞ N                                ∞ N                                
        −2n             ∗
                               dx          −2n             ∗      2         dx
          x      f (x) + f (x)     =       x        f (x) + f (x) − 2 f (1)      +
  1                             x     1                            x          x
      n =0                                     n =0
                                                                             
                                                                1          1
                                                           + 1 + + ::: +        f (1):
                                                                2        N +1
We substitute into (2.7), getting
         ∞                                
     1           Γ 1      i          1
                       + v f             + iv dv =
    2π −∞        Γ 4       2           2
                   ∞                                         
                        1 − x −2N −2            ∗      2          dx
              =−                −2
                                       f (x) + f (x) − 2 f (1)        +
                    1     1−x                         x            x
                                                                N +1
                                                                                   
                                                                    1              1
                                                  + log N −              f (1) + O     :
                                                                     n              N
                                                                        n =1

Now we take the limit for N → ∞ and deduce
               ∞                              
           1          Γ 1     i           1
                           + v f            + iv dv =
          2π −∞        Γ 4     2           2
                             ∞                           
                                             ∗      2          x dx
(2.8)                   =−         f (x) + f (x) − 2 f (1) 2          − γ f (1) =
                             1                     x          x −1
                                                ∞                           
                                                                      2           x dx
                        = −(log 4 + γ )f (1) −       f (x) + f ∗ (x) − f (1) 2         :
                                                1                     x         x   −1
If we substitute into (2.4) and note that
                              ∞                                 ∞
                     f(0) =     f ∗ (x) dx;          f(1) =         f (x) dx
                               0                                0

we get the Explicit Formula in the form stated here.
remarks on weil’s quadratic functional in the theory of prime numbers, I                191

          3. A necessary and sufficient condition for the Riemann Hypothesis
   As Weil [6] showed, one obtains from the Explicit Formula a necessary and sufficient
condition for the validity of the Riemann Hypothesis, expressed through the positivity
of a certain quadratic functional. Let us consider test functions f (x) ∈ C0∞ ((0; ∞))
which are the multiplicative convolution of a function g and its transpose conjugate g ∗ ,
hence                        ∞                        ∞
                                                 dy
                    f (x) =      g (x=y) g ∗ (y)    =     g (xy) g (y) dy:
                              0                  y     0

We have
                                   f(s) = g(s) g(1 − s):
   We have the following strengthening of Weil’s criterion.

    Theorem 1. The Riemann Hypothesis holds if and only if
                             
(3.1)                            g (ρ) g (1 − ρ) > 0
                                   ρ

for every complex-valued g (x) ∈ C0∞ ((0; ∞)), not identically 0.

    Proof. The Riemann Hypothesis is the statement that 1 − ρ = ρ for every non-trivial
zero ρ of ζ (s). Thus on the Riemann Hypothesis we have
                                                        
(3.2)               g (ρ) g (1 − ρ) =   g (ρ) g (ρ) =   |g (ρ)|2 ≥ 0:
                  ρ                      ρ                    ρ

It is also easy to show that equality holds only if g (x) is identically 0. In fact equality
can hold only if g(ρ) = 0 for every ρ, whence g(s) has at least (1=π + o(1))R log R
zeros in a disk |s | ≤ R. On the other hand, g(s) is an entire function of exponential
type, thus if g is not identically 0 it can have at most O(R) zeros in the disk |s | ≤ R.
This proves our claim about equality in (3.2) and shows that the Riemann Hypothesis
implies (3.1).
    The proof of the converse statement can be obtained by a modification of Weil’s
proof. We shall sketch here another argument in [1], based on a criterion due to X.-J.
Li [4], namely that the Riemann Hypothesis is equivalent to the statement
                                                             
                                                                    
(3.3)                    1 − (1 − 1=ρ)n + 1 − (1 − 1=(1 − ρ))n > 0
                   ρ

for n = 1; 2; 3; : : : . A quick proof is as follows [1]. Let β = (ρ) and notice that
                              |1 − 1=ρ|2 = 1 + (1 − 2β )=|ρ|2 :

This proves that (3.3) holds if (ρ) = 12 for every ρ.
    Conversely, suppose there is ρ with (ρ) < 12 . Since (1 − 2β )=|1 − ρ|2 tends to 0
as |ρ| → ∞, the maximum of this quantity as ρ varies is attained and there are finitely
many zeros ρk , k = 1; : : : ; K such that |1 − 1=ρ| = 1 + t = max > 1. For any other
192                                                                                    e. bombieri

ρ we have |1 − 1=ρ| ≤ 1 + t − δ for a fixed small δ > 0. Let φk be the argument of
1 − 1=ρk . Then
                         1 − (1 − 1=ρk )n = 1 − (1 + t )n e inφk :
                                                
For ρ = ρk we have |1 − 1=ρ|n = O (1 + t − δ )n and also
                            1 − (1 − 1=ρ)n = n=ρ + O(n2 =|ρ|2 )
as soon as |ρ| > n, as an easy calculation shows. Hence the sum over |ρ| > n is O(n2 ),
          
because      1=|ρ|2 is convergent. The number of zeros with |ρ| ≤ n is O(n2 ), again
          
because      1=|ρ|2 is convergent. Hence the contribution of terms in (3.3) other than
those arising from ρk , k = 1; : : : ; K is O n2 (1 + t − δ )n . The contribution of the
                                                 
zeros ρk , k = 1; : : : ; K is O(K ) − 2(1 + t )n cos(nφk ), and we have shown that
                                                 
                                                       
              1 − (1 − 1=ρ)n + 1 − (1 − 1=(1 − ρ))n =
       ρ
                                                   K
                                                                                              
                               = −2(1 + t )n              cos(nφk ) + O(K ) + O n2 (1 + t − δ )n :
                                                   k =1

By Dirichlet’s theorem on simultaneous diophantine approximations we can find arbi-
trarily large values of n such that the sum of cosines is arbitrarily close to K , making
it plain that the sum in (3.3) takes negative values infinitely often if the Riemann
Hypothesis does not hold.
    Now we have the identity
                                                                               
  1 − (1 − 1=s)n + 1 − (1 − 1=(1 − s))n = 1 − (1 − 1=s)n · 1 − (1 − 1=(1 − s))n :
Hence if gn (x) is the inverse Mellin transform of 1 − (1 − 1=s)n , the sum in (3.1) is
       
simply   gn (ρ) gn (1 − ρ), which, at least formally, is the left-hand side of the Explicit
Formula for gn ∗ g ∗n , because gn is real.
   The function gn is
                                   
                                    Pn (log x)
                                                  if 0 < x < 1
                           gn (x) = n=2            if x = 1
                                   
                                   
                                      0            if x > 1
where Pn (x) is the polynomial
                                            n  
                                              n   x j −1
                                 Pn (x) =                 :
                                               j (j − 1)!
                                            j =1

Since gn is not a smooth function with compact support, we cannot apply the Explicit
Formula directly. Thus we replace gn by its truncation
                                 
                                  gn (x)
                                              if ε < x ≤ ∞
                                   1
                      gn;ε (x) =   2 gn (ε)    if x = ε
                                 
                                 
                                   0           if x < ε
remarks on weil’s quadratic functional in the theory of prime numbers, I                          193


where ε > 0 and note that
                                                   
(3.4)            lim      gn;ε (ρ) gn;ε (1 − ρ) =   gn (ρ) gn (1 − ρ):
                     ε→+0
                             ρ                              ρ


This can be proved easily using a zero-free region for ζ (s), because
                                                                                      
       g (s) g (1 − s) − g (s) g (1 − s) = O εmin((s);(1−s)) (log 1=ε)n−1 =|s |2
          n;ε    n;ε          n      n

for 0 ≤ (s) ≤ 1 and |s | ≥ 1. Then using De la Vallée-Poussin’s zero-free region
                                   c                         c
                                          ≤ (ρ) ≤ 1 −
                             log(|ρ| + 2)              log(|ρ| + 2)
it is easy to verify (3.4).
                                 
     Finally, if n is such that     gn (ρ) gn (1 − ρ) < 0, by (3.4) we see that we still have

    gn;ε (ρ) gn;ε (1 − ρ) < 0 for ε > 0 small enough. The function gn;ε is real-valued
with compact support, and the proof of Theorem 1 is completed by replacing gn;ε by
a multiplicative convolution with a smooth approximation with compact support to a
Dirac at 1.
     By the Explicit Formula and Theorem 1 we have

   Theorem 2. Let T [f ] be the linear functional (3)
                     ∞              ∞               
                                                      ∞

           T [f ] =     f (x) dx +       f ∗ (x) dx −   Λ(n) {f (n) + f ∗ (n)} −
                         0                 0                n =1
                                                         ∞                     
                                                                     ∗     2        x dx
                                 − (log 4π + γ )f (1) −     f (x) + f (x) − f (1) 2
                                                         1                 x       x −1
on the space C0∞ ((0; ∞)) of complex-valued smooth functions with compact support in (0; ∞).
Then we have
                                                    
                                T [f ] = T [f ∗ ] =   f(ρ);
                                                        ρ


where the sum ranges over all complex zeros of the Riemann zeta function.
   Moreover, the Riemann Hypothesis is equivalent to the statement that
                                                   ∗
                                          T [f ∗ f ] ≥ 0

on C0∞ ((0; ∞)), with equality only if f is identically 0.

   We have in fact proved a little more, namely that the positivity of the Weil quadratic
functional is equivalent to the statement that T [gn;ε ] > 0 for every positive integer n
and ε > 0.

    (3) We use the symbol T because we regard T [f ] as an analogue of Weil’s definition of trace of a
correspondence.
194                                                                                  e. bombieri

                                     4. The variational equation
                                                        ∗
     It is easily seen that the functional T [f ∗f ] is invariant by the translation operator (4)
τa f (x) = a −1=2 f (x=a). This means however that if we try to minimize this functional in
the unit sphere of a translation invariant Hilbert space we obtain too many extremals.
Thus our Hilbert space cannot be translation invariant by τa .
     On the other hand, we want to keep the definition of the norm in our Hilbert
space as simple as possible. Hence a first attempt consists in killing the translations τa
by fixing the support of f . This works fine to some extent, and one readily shows that
           ∗
T [f ∗ f ] is bounded below in the unit sphere of the space L2 (E ) of square-integrable
functions with compact support in E , where E is a finite union of bounded closed
intervals in (0; ∞).
     However, one may also consider the infinitesimal translation given by the differential
operator D = x( d= dx), and one way to kill it is to work in the space of functions
f for which Df is square-integrable with compact support in [M −1 ; M ], because in
general Df will have a jump at M ± 1 and D 2 f will have a Dirac point mass there, so
it will no longer be square-integrable.
     These considerations motivate the two problems below.
                                              ∗
   Problem 1. Minimize T [f ∗ f ] in the unit sphere of the Hilbert space W0 =
= W01;2 ([M −1 ; M ]) of functions f with f , Df in L 2 ((0; ∞)) and compact support
in [M −1 ; M ], with norm
                                         M
                                f 2W0 =     |Df (x)|2 dx;
                                                  M −1

with D the translation invariant differential operator D = x (d=dx).
                                                                                        ∗
   Problem 2. Let E be a finite union of intervals in (0; ∞). Minimize T [f ∗ f ] in the
unit sphere of the space L2 (E ) of functions f with compact support in E , with norm
                                                
                                        f = |f (x)|2 dx:
                                           2

                                                    E

   Remark. There is some evidence that it may also be of interest to consider a variant
of Problem 1 in which instead of a Dirichlet condition f (1=M ) = f (M ) = 0 one works
with odd functions f (x) = −f ∗ (x) and imposes a Neumann condition
                             1                    1
                       (Df + f )(1=M ) = (Df + f )(M ) = 0:
                             2                    2
The analysis of this problem will not be done in this paper.
   Remark. It may be useful, in studying regularity of solutions in Problem 2, to
consider also viscosity solutions, namely minimizers of the regularized functional T [f ∗
 ∗
f ] + ε Df 2 for ε > 0, and their limits in L2 (E ) as ε → 0.

      (4) We view this as the analogue of a normalized Frobenius operator.
remarks on weil’s quadratic functional in the theory of prime numbers, I                                     195


   Lemma 1. The variational equation for Problem 1 is
(4.1)                  λDf (a) − L[f ](a) = 0                   for a ∈ (M −1 ; M )
where D is the Laplacian D = −D − D 2 and L[f ] is the Euler-Lagrange linear operator
                     ∞               ∞               ∞                        
                                           1     a                         1    a
        L[f ](a) =       f (ay) dy +         f     dy −       Λ(n) f (an) + f         −
                      0                0   y     y                         n    n
                                                         n =1
                                                ∞                          
                                                              1   a     2        x dx
                        − (log 4π + γ )f (a) −      f (ax) + f        − f (a)    2
                                                                                      :
                                                1             x   x     x      x   −1
   The eigenvalue λ is given by
                                                                    ∗
                                                    T [f ∗ f ]
                                          λ=                        2
                                                                        :
                                                        Df
   The variational equation for Problem 2 is
(4.2)                        λ f (a) − L[f ](a) = 0                     for a ∈ E
and the eigenvalue λ is given by
                                                                    ∗
                                                    T [f ∗ f ]
                                          λ=                    2
                                                                        :
                                                            f
   Proof. Let                                    ∞
                                       u; v =               u(x) v(x) dx
                                                    0

be the scalar product in L2 ((0; ∞)), so that v; u = u; v . Let us make a variation
f + εϕ of f , where ϕ is a complex-valued smooth function with compact support. We
have
                                   ∗                    ∗
        T [(f + εϕ) ∗ (f + εϕ) ] = T [f ∗ f ] + 2 ε [ L[f ]; ϕ ] + ε2 T [ϕ ∗ ϕ∗ ]
and
                   D(f + εϕ) 2 = Df             2
                                                    + 2 ε [ Df; ϕ ] + ε2 D ϕ 2 :
                                                                            ∗       2
Hence the vanishing of the first variation of T [f ∗ f ]= Df                            gives the equation
                               [ L[f ]; ϕ − λ Df; ϕ ] = 0
for every complex-valued smooth ϕ with compact support in (0; ∞), with λ = T [f ∗
  ∗
f ]= Df 2 . This completes the proof for the first assertion of the lemma. The proof
of the second assertion is identical, mutatis mutandis.
    We shall prove that the infimum for the preceding two variational problems is
attained. We need some simple preliminary results.
   Lemma 2. If f ∈ W0 then f is Hölder continuous with exponent 1=2, and verifies the
pointwise inequality
                                                     1=2
                                                   
                        f (y) − f (x) ≤ Df  1 − 1 
                                               x   y
196                                                                                            e. bombieri

for 0 < x < y < ∞. In particular, f is bounded as |f (x)| ≤ M 1=2 Df                          and           f   ≤
≤ (2 log M )1=2 Df .
   Moreover, for any f; g ∈ L2 ((0; ∞)) we have the pointwise bound
                                                            
                                               (f ∗ g ∗ )(x) ≤ f √· g :
                                                                     x
      Proof. We have                                             y
                                                                             dt
                                              f (y) − f (x) =      (Df )(t )
                                                                 x           t
and the first statement of the lemma follows from Cauchy’s inequality.
   The second statement is also clear, because
                                            ∞
                                   ∗
                             (f ∗ g )(x) =     f (xy) g (y) dy
                                                                0
                2           −1       2
and        |f (xy)| dy = x        f       .
      Lemma 3. For f ∈ L2 with compact support in [M −1 ; M ] we have
                         ∞                               
                ∗     1         Γ 1      i         1         2
        T [f ∗ f ] =                  + v f          + iv  dv + O(M ) f                         2
                                                                                                        :
                     2π −∞       Γ 4      2          2
The constant involved in the O(                 ) symbol is absolute.
                                                                                          ∗
  Proof. Since f is supported in [M −1 ; M ] we see that F = f ∗ f is supported in
[M ; M 2 ]. Hence by the last estimate of Lemma 2 we have
      −2


             ∞                           M2
                      dx   ∞                       dx
                F (x)       =     F (x) dx ≤       f 2 √ < 2M f 2 :
                      x                      − 2      x
              0                  0             M

      Again by Lemma 2, we get in a similar way
                   ∞              
                                          M2
                                                 Λ(n)
                                         2                                      2
                  2      Λ(n)F (n) ≤ 2 f        √ = O(M ) f                         :
                                                  n
                     n =1                    n =1

      Also,
                                                                     2
                                                      F (1) = f          :
                                                                                  ∗
      Finally, by the Explicit Formula the last two terms in T [f ∗ f ] are
                                    ∞                              
                                 1         Γ 1      i         1        2
               −(log π )F (1) +                  + v f         + iv  dv:
                                2π −∞      Γ 4       2          2
This proves Lemma 3.
   Lemma 4. Let f ∈ L2 be with compact support in [M −1 ; M ]. Then we have the pointwise
bound                
                    d k                
                                1         √
                           f     + it  ≤ 2 (log M )k +1=2 f :
                    dt          2        
remarks on weil’s quadratic functional in the theory of prime numbers, I                             197


   Proof. We have
                       M                                    
        d k                                                         
                 1                                      1
              f   + it  =  i k     f (x) (log x)k x − 2 +it dx  ≤
        dt       2                1=M
                                                     M                              M       12
                                                k              |f (x)|                    dx
                                   ≤ (log M )                    √ dx ≤ (log M )k                    f :
                                                        1=M        x                  1=M x

This proves the lemma.

    Theorem 3. Let E be a finite union of closed finite intervals in (0; ∞). Then the infimum
             ∗
of T [f ∗ f ] in the unit sphere of the space L2 (E ) of L2 -functions with compact support in E
is attained.
                                                                                                      ∗
   Proof. Let {fν } be a minimizing sequence for this problem and let λ = inf T [f ∗ f ].
Let E ⊂ [M −1 ; M ].
   For 0 < (w) we have
                                   Γ   w
(4.3)                                              = log |w | + O(1)
                                   Γ    2
and
                                   ∞         2
                                                
                              1      f 1 + iv  dv = f                 2
                             2π         2       
                                   −∞

by Plancherel’s formula. Hence Lemma 3 shows that
                                            ∗
                                   T [fν ∗ f ν ] ≥ −C1 M fν 2
                                                    ∗
and in particular the functional T [f ∗ f ] is bounded below in L2 (E ). Since this
functional is automatically bounded above along a minimizing sequence, Lemma 3 and
(4.3) also shows that
                      ∞                             2
                                               1       
(4.4)                     (1 + log+ |v |) fν   + iv  dv = O(1)
                      −∞                        2
uniformly in ν as ν → ∞.
   By Lemma 4, the coefficients aν ;m of the Taylor expansion with center 12 + it of
the entire function fν (s) are bounded by C (δ )δ m , m = 0; 1; : : : , for any δ > 0, for
a suitable constant C (δ ) depending only on δ . Hence by a standard diagonal selection
process there is a subsequence of the functions fν , which we still denote by {fν } for
simplicity, such that the limits
                                         lim aν ;m = am
                                        ν →∞

exist and again |am | ≤ C (δ )δ m . Therefore, setting
                                                     
                                                     ∞

                                        f(s) =               am s m
                                                     m =0
198                                                                                         e. bombieri

we see that f(s) is an entire function and that the sequence {fν (s)} converges to f(s)
uniformly on compact subsets of C. It follows that for any fixed T we have
                          T                          2
                                   1              1       
                                                        
(4.5)                lim
                    ν →∞      f ν 2 + iv − f 2 + iv  dv = 0:
                          −T

For the remaining integral over |v | > T , we note that by lower semicontinuity we have
       ∞                           2              ∞                            2
                              1                                              1       
(4.6)      (1 + log+ |v |)f   + iv  dv ≤ lim inf     (1 + log+ |v |) fν   + iv  dv:
        −∞                     2               ν →∞
                                                       −∞                       2
In conjunction with (4.4), this proves
                                         2             
                                              
                       f 1 + iv − f 1 + iv  dv = O     1
                        ν 2             2              log T
                |v |>T

and by (4.5) it follows that
                         ∞                         2
                                  1             1       
                    lim      
                             
                                
                               f ν   + iv − f     + iv  dv = 0:
                   ν →∞
                         −∞        2             2
Therefore, by Plancherel’s formula we get limν fν − f                    = 0 and the sequence {fν }
converges to f strongly in L2 (E ).
   Finally, we have
                                                  ∗              ∗
                                   lim T [fν ∗ f ν ] = T [f ∗ f ]:
                                  ν →∞
                                                                     ∗                  ∗
This is verified as follows. Let us abbreviate Fν = fν ∗ f ν and F = f ∗ f . Then by
Lemma 2 we see that
                                                   ∞                 
                 ∞                                                   
 F (x) − F (x) =           f   (xy) f   (y) dy −     f (xy) f (y) dy ≤
   ν                           ν        ν                              
                          0                          0
                     ∞                                    ∞                                
                                                                                      
                 ≤          fν (xy) − f (xy) fν (y) dy  +       f (xy) fν (y) − f (y) dy  ≤
                        0                                         0
                             ∞                            12                ∞                12
                 ≤ fν                |fν (xy) − f (xy)|2 dy + fν − f                 |f (xy)|2 dy     =
                             0                                                0
                                                                                      2
                                                                                   = √ fν − f :
                                                                                       x
Since {fν } converges strongly to f in L2 , this proves that {Fν (x)} converges uniformly
to F (x) as ν → ∞. This suffices to show that in the right-hand side of the Explicit
                      ∗                                                                ∗
Formula for T [fν ∗ f ν ] all terms converge to the corresponding terms for T [f ∗ f ],
except possibly for the last term. However, by (4.3), (4.5) and lower semicontinuity
we have
             ∞                                2
                    Γ 1        i          1       
                           + v f          + iv  dv ≤
             −∞     Γ     4    2            2
                                              ∞                              2
                                          1           Γ 1    i          1        
                               ≤ lim inf                  + v fν         + iv  dv:
                                  ν →∞ 2π             Γ 4     2           2
                                              −∞
remarks on weil’s quadratic functional in the theory of prime numbers, I                  199


Therefore, we obtain
                                            ∗                      ∗
                                  T [f ∗ f ] ≤ lim T [fν ∗ f ν ]
                                                    ν →∞

and our assertion follows because {fν } is a minimizing sequence.
   This completes the proof of Theorem 3.

   We can prove the corresponding result for Problem 1 only in the hypothetical case
                 ∗
in which T [f ∗ f ] is negative for some f ∈ W0 .
                                                               ∗
  Theorem 4. Let λ be the infimum of T [f ∗ f ] in the unit sphere of the space W0 =
= W01;2 ([M −1 ; M ]).
  If λ < 0, then this infimum is attained.

   Proof. The proof follows the same pattern as for Theorem 3. Let again {fν } be a
minimizing sequence and f a weak limit in W0 of a subsequence of {fν }.
   This time, the normalization Dfν = 1 means that
                        ∞                      2
                     1         1            1       
                                 + v 2 fν   + iv  dv = 1;
                    2π −∞ 4                  2
                              ∗                 ∗
and the proof that T [fν ∗ f ν ] → T [f ∗ f ] is immediate.
    The trouble here is that there is no immediate reason why Df should continue
to have norm 1, and we have only Df ≤ 1 by semicontinuity.
                                                      ∗
    However, suppose that λ < 0. Then T [f ∗ f ] = λ < 0 and in particular f
cannot be identically 0. Thus 0 < Df ≤ 1. Now f0 = f= Df has Df0 = 1 and
          ∗
T [f0 ∗ f 0 ] = λ= Df 2 . If we had Df < 1 then the assumption λ < 0 shows that
                                                                     ∗
λ= Df 2 < λ, contradicting the fact that λ is the infimum of T [f ∗ f ] in W0 . It
follows that Df = 1 and, by well-known results, fν converges strongly to f in W0 ,
proving the theorem.
    We conclude this section with

    Theorem 5. Let µ+ (M ) and µ− (M ) be the infimum of T [f ∗ f ∗ ] in the class of even
and odd functions in L2 ([M −1 ; M ]) of norm 1. Then µ+ (M ) and µ− (M ) are continuous
decreasing functions of M .

    Proof. It is clear that the functions µ± (M ) are decreasing functions of M , so we
need to prove continuity.
    Let f (x) be even or odd and a minimizer for T [f ∗ f ∗ ]. Consider, for ε > 0, the
variation
                                                           ε
                                  fε (x) = (1 + ε)1=2 x 2 f (x 1+ε ):
Then fε (x) ∈ L2 ([M −1=(1+ε) ; M 1=(1+ε) ]), has norm 1 and has the same parity as f .
  It follows that, with the sign ± according to the parity of f :
             lim inf µ± (M 1=(1+ε) ) ≤ lim inf T [fε ∗ fε∗ ] = T [f ∗ f ∗ ] = µ± (M ):
              ε→0                         ε→0
200                                                                             e. bombieri

Here the equality lim T [fε ∗ fε∗ ] = T [f ∗ f ∗ ] can be verified directly without much
trouble, because fε converges strongly in L2 ([M −1 ; M ]) to f and because
                                                              
                                        − 12            ε
                        f ε (s) = (1 + ε) f s +                  ;
                                                      2(1 + ε)
this formula being used to show that
                         ∞                          2
                                               1       
                                     +      
                             (1 + log |v |)f ε   + iv  dv
                          −∞                    2
is a continuous function of ε as ε → 0. This completes the proof.

                              5. The first eigenvalue problem

    In this section we do a formal calculation which will suggest the structure of solutions
of the eigenvalue problem of the preceding sections.
    We assume a priori throughout this section that we have f (x) ∈ W0 such that
(5.1)                     λDf (x) = L[f ](x);          x ∈ (M −1 ; M ):

   Lemma 5. The distribution D 2 f is the sum of a bounded function and two Dirac
                                                                              distributions
                                                                                         
at x = M ±1 . Hence the Mellin transform f(s) satisfies a bound f(s) = O (|s | + 1)−2 in
any fixed vertical strip a ≤ (s) ≤ b. The constant implied in the O( ) symbol depends on M
and a; b.

      We also have
                                             1
                      L[f ](x) = O(M )x − 2 Df          for x ∈ (M −1 ; M ):
                                                                                    1
   Proof. Consider L[f ](x) for x ∈ (M −1 ; M ). By Lemma 2, |f (x)| ≤ x − 2 Df .
For x ∈ (M −1 ; M ), the functions f (xy) and f (x=y) are supported in y ∈ (M −2 ; M 2 ).
Hence
                  ∞               ∞       
                                       1    x                  1
                      f (xy) dy +        f       dy = O(M )x − 2 Df :
                   0               0   y    y
      The next term in L[f ] is
                                  
                                  ∞                              
                                                      1       x
                              −          Λ(n) f (nx) + f
                                                      n       n
                                  n =1
                                                                          1
and, by the same argument as before, it is majorized by O(M )x − 2 Df .
                                                   1
   Also it is clear that −(log π )f (x) = O(x − 2 Df ). The remaining contribution to
L[f ] is
                               
                            1           Γ w
                                                 x −w f(w) dw:
                          2π i ( 1 )     Γ 2
                                   2
remarks on weil’s quadratic functional in the theory of prime numbers, I                  201


Since f ∈ W 1;2 we have by Plancherel’s formula
                          
                       1
                                 |w |2 |f(w)|2 dw = Df             2
                                                                        < ∞;
                      2π i ( 1 )
                                  2

and since Γ =Γ = O(log |w |) we see from Cauchy’s inequality that the contribution
coming from this last term is again O(x −1=2 Df ). This proves the last statement of
Lemma 5.
   By (5.1), we see that
                            Df (x) = −(D + D 2 )f (x) = −(x 2 f  (x))
is bounded in (M −1 ; M ). It follows that f (x) is of differentiability class C 1;1 in
(M −1 ; M ). Since Df is 0 outside [M −1 ; M ] we see that Df (x) has right and left
limits for x → M ±1 and the first statement of the lemma follows. The second statement
now is clear from                         ∞
                             s 2 f(s) =     D 2 f (x)x s −1 dx:
                                              0

This completes the proof.
   Lemma 6. For a solution f (x) we have
                                                   
                                      L[f ](x) =        f(ρ)x −ρ
where the sum ranges over all non-trivial zeros of ζ (s). In particular, the sum is absolutely
convergent because f(ρ) = O(|ρ|−2 ).
    Proof. Since L[f ](a) = T [f (ax)] and the Mellin transform of f (ax) is a −s f(s),
Lemma 6 follows from Theorem 2 and Lemma 5.
    We write the equation λDf = L[f ] in the form
                                                
(5.2)                        −λ(x 2 f  (x)) =   f(ρ)x −ρ
                                                        ρ

             −1
for x ∈ (M        ; M ), with the boundary condition f (M ) = f (1=M ) = 0.
   Lemma 7. Suppose λ = 0. Then for x ∈ (1=M; M ) we have
                                              f(ρ) −ρ
                             λ f (x) =                x − A − Bx −1
                                             ρ(1 − ρ)
                                         ρ

where the constants A and B are determined by the boundary condition f (1=M ) = f (M ) = 0.
    Moreover, for every α < 1 the function f is of differentiability class C 2;α in the open
interval (M −1 ; M ).
   Proof. We recall that by Lemma 3 we have f(ρ) = O(|ρ|−2 ), where the constant
implied in the O( ) symbol depends only on M . It follows that the function
                                                        f(ρ) −ρ
                                 f0 (x) = λ−1                   x
                                                       ρ(1 − ρ)
                                                   ρ
202                                                                                            e. bombieri

is for α < 1 a C 2;α ((0; ∞)) solution of
                                          
                       −λ(x 2 f0 (x)) =   f(ρ)x −ρ ;               x ∈ (0; ∞);
                                                      ρ

as one verifies directly.
    The associated homogeneous equation −(x 2 u  ) = 0 has general solution u(x) = A +
+ B=x. Thus we have f (x) = f0 (x) + u(x), and A, B are determined by the boundary
condition. This completes the proof of the lemma.

                                       6. The dual eigenvalue problem
    Let φ(x) be the characteristic function of (1=M; M ). Consider complex-valued
functions f ∈ W0 given by
                  
(6.1)     f (x) =   Xρ φ(x)x −ρ + X0 φ(x) + X1 φ(x)x −1 if x ∈ (M −1 ; M )
                         ρ


                   = (M −1 ; M ). Here the sum runs over all complex zeros ρ of ζ (s),
and f (x) = 0 if x ∈
repeated according to their multiplicity m(ρ).
   The coefficients X0 and X1 are determined by the condition f (M ) = f (1=M ) = 0.
An easy calculation shows that
                                                 M 1−ρ − M ρ−1
                                       X0 = −                           Xρ ;
                                                  ρ
                                                           M − M −1
(6.2)
                                                 M ρ − M −ρ
                                       X1 = −                        Xρ :
                                                  ρ
                                                          M − M −1

      We have
                                       
(6.3)                        f(s) =        Xρ φ(s − ρ) + X0 φ(s) + X1 φ(s − 1):
                                        ρ

By Lemma 7, for f (x) to be a formal solution of our eigenvalue problem it will be
sufficient (5)
                                        f(ρ)
                                Xρ =            :
                                     λρ(1 − ρ)
      In view of (6.3), this yields the eigenvalue equation
            λρ(1 − ρ)Xρ =
                                                                  
(6.4)                             M 1−ρ − M ρ −1         M ρ − M −ρ 
               =             
                      φ (ρ − ρ ) −                φ (ρ ) −             φ (ρ − 1) Xρ :
                  ρ
                                     M − M −1               M − M −1


      (5) This condition is only sufficient because there may be lack of uniqueness in the expansion (6.1).
remarks on weil’s quadratic functional in the theory of prime numbers, I                  203

                                     7. The resolvent
   We investigate here directly the linear system (6.4). We make a change of variables
by setting ρ = 12 + i γ , M = e t with t > 0, Λ = 1=λ and defining the multiplicity m(γ )
of γ as m(γ ) = m(ρ). We also abbreviate
                                                  sin(x)
                                       K (x) =
                                                     x
and
                                                                 
                                  t      1           i           i
   K ∗ (x; y; t ) = K t (x − y) −            + iy K t    −y   K t     +x    −
                                 sinh(t ) 2            2           2
                                                                 
                                    t      1           i            i
                               −             − iy K t    +y   K t     −x    :
                                 sinh(t ) 2            2           2
It is easily verified that
                                           
          1                           1
             + x 2 K ∗ (x; y; t ) =      + y 2 K ∗ (y; x; t ) =
          4                           4
(7.1)                                                                               
                          1                     cosh(t ) cos t (x − y) − cos t (x + y)
                     =      + xy K t (x − y) −                                             :
                          4                                       2t sinh(t )
    After the change of variables and setting
                                                 
                                           1    2
(7.2)                zγ = Xρ ; wγ =          + γ zγ ;                Λ = 1=λ
                                           4
and
                                                 2t K ∗ (x; y; t )
(7.3)                           H (x; y; t ) =
                                                    ( 14 + y 2 )
equation (6.4) becomes
                                        
(7.4)                          wγ = Λ         H (γ ; γ  ; t ) wγ  :
                                         γ

   We write
                              H(Γ; t ) = H (γ ; γ  ; t )
                                                              γ ;γ  ∈ Γ

and D(Λ; t ) for the resolvent determinant
                                                         
(7.5)                         D(Λ; t ) = det I − ΛH(Γ; t ) :
We have
    Theorem 6. The resolvent (7:5) is well defined and is an entire function of the complex
variable Λ, at most of order 1 and finite exponential type.
    Moreover,
                                             ∞
                                                               Λn
                             D(Λ; t ) = 1 +      (−1)n ∆n (t )
                                                               n!
                                               n =1
204                                                                                             e. bombieri

where
                                        
                        ∆n (t ) =                    det H (γj ; γk ; t )                   :
                                    γ1 ;::: ;γn ∈Γ                          j;k =1;::: ;n

      Finally, as N → ∞ the truncations DN (Λ; t ) given by

                                 det δγ ;γ  − Λ H (γ ; γ  ; t )
                                                                       γ ;γ  ∈ΓN
                
with ΓN = Γ         {|γ | ≤ N }, converge to D(Λ; t ) uniformly for Λ in compact subsets of C.
      Proof. We claim that
                                                      
(7.6)                                  H (γ ; γ  ; t ) <+ ∞:
                                      γ ;γ 

To see this, we start with the inequality
                                                                         
                                              1                     1
(7.7)                 H (x; y; t )                       min 1;
                                     (1 + |x |)(1 + |y |)        |x − y |
valid for fixed t and x, y in a fixed horizontal strip excluding a neighborhood of
x = ±i=2 (this last restriction is inconsequential in what follows). Thus (7.6) is implied
by the convergence of the series
                                           1               1
                                                                      :
                           γ ;γ 
                                  (1 + |γ |)(1 + |γ |) 1 + |γ − γ  |
                                                   



   We split the summation over γ  into subsums where γ  ranges over n ≤ |γ  − γ | <
< n + 1, for n = 0; 1; 2; : : : . Hence the series is majorized by
                                     1 1
                                               ∞
                                                                          1
(7.8)                                                                              :
                                   1 + |γ | n                           1 + |γ  |
                             γ                 n =1     n−1≤|γ  −γ |<n

   In (7.8), we split the sum over n into the three ranges n ≤ [|γ |=2], [|γ |=2] < n <
< [3|γ |=2] and n ≥ [3|γ |=2]. In the first range, γ  is of order γ and (6)
                                  
                                           1  log(|γ | + 2):
                                    n≤|γ  −γ |<n+1

Thus the sum over the first range contributes to (7.8), up to a constant factor, an
amount
            1  [|    γ |=2]
                               log(|γ | + 2)
                                              
                                                  (log(|γ | + 2))2
                                                                  <+ ∞:
           γ
             1 + |γ |
                        n =1
                                (1 + |γ |)n     γ
                                                     (1 + |γ |)2
   The contribution of the sum over the third range is computed in a similar way. This
time γ  is of precise order n and the number of points γ  in the range n −1 ≤ |γ  −γ | < n

    (6) We use Vinogradov’s notation a  b to indicate an inequality |a | ≤ Cb for some unspecified
positive constant C .
remarks on weil’s quadratic functional in the theory of prime numbers, I                                     205


is O(log(n + 2)). Thus the sum over the third range contributes to (7.8), up to a
constant factor, an amount
              1   log(n + 2)   log(|γ | + 2)
                                                             <+ ∞:
              γ
                 1 + |γ |
                          n>[3|γ |=2]
                                      n2       γ
                                                  (1 + |γ |)2
                                                                                
   In the second range |γ  |, up to a constant factor, is at least 1 + n − |γ | and the
number of points γ  in the range n − 1 ≤ |γ  − γ | < n is O(log(|γ | + 2)). Thus the
sum over the second range contributes to (7.8), up to a constant factor, an amount
          log(|γ | + 2)                                    1          log(|γ | + 2)2
                                                                                     <+ ∞:
          γ
              (1 + |γ |)2       [|γ |=2]<n<[3|γ |=2]
                                                       1 + n − |γ |      (1 + |γ |)2
                                                                              γ


This completes the proof of (7.8) and, with it, of (7.6).
   Once we have (7.6), the convergence of DN (Λ; t ) to a limit is an application of von
Koch’s criterion [7, Ch. II, §2.81, p. 36] for the convergence of an infinite determinant.
   We have a Taylor expansion
                                                           J +1
                                                                                   Λn
                                 DN (Λ; t ) = 1 +                 (−1)n ∆n;N (t )
                                                                                    n!
                                                           n =1

where J + 1 is the cardinality of ΓN and
                                              
(7.9)                 ∆n;N (t ) =                         det H (γj ; γk ; t )                      :
                                        γ1 ;::: ;γn ∈ΓN                             j;k =1;::: ;n


Now, using (7.7), we apply Lord Kelvin’s inequality (also more widely known as
Hadamard’s inequality) to each determinant and find
               n
                                    n
                                     
                                                               1                         1
(7.10)           H (γ ; γ ; t )2                                                                      :
                            j      k                                        2             2
               k =1                               k =1
                                                       1 + ( |γj | − |γk |)   (1 + |γj |)   (1 + |γk |)2
    Since the number of γk in an interval (m; m + 1) is O(log(m + 2)), the right-hand
side of (7.10) is majorized, up to a constant factor, by
                          1        
                                   ∞
                                         log(m + 2)       log(|γj | + 2)
                                                    2                 :
                                 2     2            
                      (1 + |γj |) m=1 m (1 + m − |γ | )    (1 + |γj |)4
   Thus the application of Lord Kelvin’s inequality yields
                                                             n                
                                                            n
                                                                      log(|γj | + 2) 1=2
              det H (γj ; γk ; t )                  ≤ A(t )
                                      j;k =1;::: ;n
                                                                j =1
                                                                       (1 + |γj |)4

for some constant A(t ) depending only on t . We also have
                          n                                     
                                log(|γj | + 2) 1=2        log(|γ | + 2) n
                                                   =                      :
               γ ;::: ;γn j =1
                                 (1 + |γj |)4         γ
                                                         (1 + |γ |)2
                  1
206                                                                                                           e. bombieri

      This proves that
                                                                           n
                                             |∆n;N (t )| ≤ A(t )B
          
with B =        log(|γ | + 2) (1 + |γ |)−2 <+ ∞. This bound, uniform in N , continues
to hold for ∆n (t ).
   It is now obvious that D(Λ; t ) is an entire function of Λ, at most of order 1 and
exponential type bounded by log(A(t )B).
   This completes the proof of Theorem 6.

   Our next result shows that zeros of the resolvent give rise to solutions f ∈ W0 of
the eigenvalue problem λDf = L[f ] for x ∈ (1=M; M ).

      Theorem 7. Let Λ0 be a zero of D(Λ; t ) = 0, of multiplicity m. Define
                                                           
                                                           ∞
                                                                                            Λn
                          D(γ ; γ0 ; Λ; t ) = Λ                   (−1)n ∆n (γ ; γ0 ; t )
                                                                                            n!
                                                           n =0

where ∆0 (γ ; γ0 ) = H (γ ; γ0 ; t ) and in general
                                                        H (γ ; γ0 ; t )         H (γ ; γk ; t )
            ∆n (γ ; γ0 ; t ) =                    det                                                             :
                                 γ1 ;::: ;γn ∈Γ          H (γj ; γ0 ; t )        H (γj ; γk ; t ) j;k =1;::: ;n

Then D(γ ; γ0 ; Λ; t ) is an entire function of Λ, at most of order 1 and finite exponential type.
   There exist γ0 and an integer l , 0 ≤ l ≤ m − 1, such that the vector {wγ } with components
                                         1
                                     wγ =  (@=@ Λ)l D(γ ; γ0 ; Λ0 ; t )
                                        l!
is a solution, not identically 0, of the linear system (7:6).
     We also have
                                        
                                wγ  log(|γ | + 2)=(1 + |γ |)2 :
  Moreover, the function f (x) defined by (6:1), (6:2), M = e t and Xρ = zγ is a solution in
W0 of
                                 Df (x) = Λ0 L[f ](x);                x ∈ (1=M; M )
with Dirichlet boundary conditions f (1=M ) = f (M ) = 0.

      Proof. This is obtained (see the discussion in the continuous case in [7, Ch. XI,
§11.23, pp. 219-220]) by solving first the finite linear system (7.4) restricted to ΓN and
then going to the limit as N → ∞.
   Let J + 1 be the number of points γ ∈ ΓN and define
                                                          H (γ ; γ0 ; t )         H (γ ; γk ; t )
          ∆n;N (γ ; γ0 ; t ) =                     det                                                                :
                                 γ1 ;::: ;γn ∈ΓN           H (γj ; γ0 ; t )        H (γj ; γk ; t ) j;k =1;::: ;n
remarks on weil’s quadratic functional in the theory of prime numbers, I                          207


   Then we set
                                                J
                                                                                    Λn
                      DN (γ ; γ0 ; Λ; t ) = Λ          (−1)n ∆n;N (γ ; γ0 ; t )         :
                                                                                     n!
                                                n =0

   Consider the (J + 1) × (J + 1) matrix

                                δjk − Λ H (γj ; γk ; t )
                                                                  j;k =0;1;::: ;J

where δjk is Kronecker’s delta. Then for j = 1; : : : ; J the cofactor of the (j + 1)-th
element of the first row of this matrix turns out to be DN (γ ; γ0 ; Λ; t ). If j = 0 the
cofactor is given by a modified formula, namely
                                                                                       Λ J +1
                DN (γ0 ; γ0 ; Λ; t ) + DN (Λ; t ) − (−1)J +1 ∆J +1;N (t )                     :
                                                                                     (J + 1)!
    By Laplace’s identity, it follows that if
                                                                                
                                                                          Λ J +1
(7.11) Wγ ;N = DN (γ ; γ0 ; Λ; t ) + DN (Λ; t ) − (−1)J +1 ∆J +1;N (t )            δ   ;
                                                                        (J + 1)! γ ;γ0
where δγ ;γ0 is Kronecker’s delta, then
                                                        
(7.12)               Wγ ;N = ΛDN (Λ; t ) + Λ                      H (γ ; γ  ; t )Wγ  ;N :
                                                       γ  ∈Γ N

    Now we let N → ∞. By Theorem 5, DN (Λ; t ) converges uniformly in compact
subsets to the entire function D(Λ; t ). Moreover, the same proof as in Theorem 5
shows that DN (γ ; γ0 ; Λ; t ) → D(γ ; γ0 ; Λ; t ), an entire function of Λ of order at most
1 and finite exponential type. Thus we can pass to the limit as N → ∞ in the above
equation, and notationwise its effect is to drop the suffix N . In the limit as N → ∞, we
modify the definition of Wγ0 by omitting the term −(−1)J +1 ∆J +1;N (t )ΛJ +1 =(J + 1)!,
because it tends to 0 as N → ∞.
    We have the Fredholm identity
                                                       @D(Λ; t )
(7.13)                          D(γ ; γ ; Λ; t ) = −Λ             :
                              γ
                                                           @Λ

Since D(Λ; t ) has a zero of multiplicity m at Λ = Λ0 , the above formula shows that
there is γ0 such that D(γ0 ; γ0 ; Λ; t ) has a zero of multiplicity exactly l ≤ m − 1 at
Λ = Λ0 and every D(γ ; γ0 ; Λ; t ) vanishes at Λ = Λ0 with multiplicity not less than l .
By differentiating (7.13) l times and setting
                                        1 @ l
                                 wγ =      ( ) D(γ ; γ0 ; Λ0 ; t )
                                        l ! @Λ
we get
                                            
                                 wγ = Λ0             H (γ ; γ  ; t ) wγ
                                            γ  ∈Γ

and wγ0 = 0.
208                                                                                          e. bombieri

    The application of Lord Kelvin’s inequality shows that
                                                     
                        1 @ l                          log(|γ | + 2)
                           ( ) D(γ ; γ0 ; Λ0 ; t )                  :
                        l ! @Λ                        (1 + |γ |)2
                                
This proves that Xρ = zγ  log(|γ | + 2)=(1 + |γ |)4 ; thus the series for f (x) and
Df (x) are both absolutely convergent. Since f (1=M ) = f (M ) = 0 is assured by the
structure of the linear system (6.4), we conclude that f ∈ W0 and with it the proof of
Theorem 7.

                          8. The eigenvalues of finite approximations
    In this section we analyse more closely the case in which the set of points ρ is a
finite set with the same symmetries as the set of zeros of the Riemann zeta function.
The main result of this section, namely Theorem 8, shows that the matrix H (γ ; γ  ; t )
has only real eigenvalues and determines the number of negative eigenvalues and the
multiplicity of 0 as an eigenvalue.
    It proves to be notationally convenient to state all our results in terms of the complex
Fourier transform rather than the Mellin transform.
    Let Z be a finite multiset (7) of complex numbers, repeated according to their
multiplicity. We assume that if ρ ∈ Z then ρ and 1 − ρ are again elements of Z , with
the same multiplicity as ρ. For ρ ∈ Z we write ρ = 12 + i γ and denote by Γ the
corresponding set of points γ .
    We denote by Φ(x) the characteristic function of the closed interval [−t; t ] and let
V be the C-vector space generated by linear combinations of the functions Φ(x)e x=2 ,
Φ(x)e −x=2 and Φ(x)e −i γ x for γ ∈ Γ. We also denote by V ◦ the subspace of codimension
2 consisting of functions F (x) ∈ V satisfying the additional condition
                                          F (−t ) = F (t ) = 0:
Finally by (P; Q ) we denote the usual inner product in L2 (R) given by
                                        ∞
                             (P; Q ) =      P (x) Q (x) dx:
                                                   −∞

    Consider the linear functional on V given by (8)
                                      ∞
(8.1)                     L[F ](x) =        F (x + u) e i γ u du:
                                            γ ∈Γ   −∞


We have

      Lemma 8. If F ∈ V then the inner product (L[F ]; F ) is real.

    (7) By a multiset we mean a set whose elements have positive integral multiplicity.
    (8) Here and in what follows the sum is understood to be over the multiset Γ, so that each term appears
according to its proper multiplicity.
remarks on weil’s quadratic functional in the theory of prime numbers, I                        209


    Proof. For F ∈ V we have, using Γ = −Γ,
                                    ∞  ∞                      
                                                     i γ (u −x)
                     (L[F ]; F ) =           F (u) e            du F (x) dx =
                                         −∞     γ ∈Γ   −∞

                                         ∞  ∞
                                    =                    F (u)F (x) e i γ (u−x) dx du =
                                        γ ∈Γ   −∞   −∞

                                         ∞  ∞
                                    =                    F (u)F (x) e i γ (x −u) dx du =
                                        γ ∈Γ   −∞   −∞

                                         ∞  ∞
                                    =                    F (u)F (x) e −i γ (u−x) dx du =
                                        γ ∈Γ   −∞   −∞

                                         ∞  ∞
                                    =                    F (u)F (x) e i γ (u−x) dx du = (L[F ]; F ):
                                        γ ∈Γ   −∞   −∞


All steps are justified by Fubini’s theorem, because F is bounded with compact support,
concluding the proof.
    Let ∆ = −(d=dx)2 . If P (x) is a smooth function on [−t; t ] and P (−t ) = P (t ) = 0
we have by integration by parts
                             t                    t
                                                        2
(8.2)                           ∆P (x) P (x) dx =     P (x) dx:
                             −t                        −t

    A first consequence is the following result.
    Lemma 9. Suppose that F (x) ∈ V ◦ is not identically 0 and satisfies the eigenvalue equation
                               
                         1
(8.3)               λ      + ∆ F (x) = L[F ](x) for − t < x < t:
                         4
    Then λ is real.
    Proof. We take the inner product of (8.3) with F . Since F ∈ V ◦ we verify by
integration by parts that
                                                ∞
                1                1 ∞          2
                   + ∆ F; F =          |F (x)| dx +     |F  (x)|2 dx > 0:
                4                4 −∞                −∞

The lemma follows because (L[F ]; F ) is real.
    Corollary. With the notation of the preceding section, the resolvent
                                                               
(8.4)                      DN (Λ; t ) = det I − Λ H(ΓN ; t )
has only real roots.

   Lemma 10. The matrix H(Γ; t ) admits the eigenvalue 0 if and only if there is γ ∈ Γ
with multiplicity greater than 1, in which case the multiplicity of 0 as an eigenvalue is exactly
                      
   [m(γ ) − 1], where        ranges over all distinct γ ∈ Γ.
   Moreover if every γ is real all eigenvalues of H(Γ; t ) are non-negative.
210                                                                                                       e. bombieri

     Proof. We have
                        t                       1              1           1              1
    1                                          e ( 2 −iy)t − e −( 2 −iy)t e ( 2 +ix)t − e −( 2 +ix)t
         2
      + y H (x; y; t ) =     e ixu e −iyu du −                                                       −
    4                     −t                           e t − e −t                   1
                                                                                    2 + ix

                                                                     1               1          1                 1
                                                                   e ( 2 +iy)t − e −( 2 +iy)t e ( 2 −ix)t − e −( 2 −ix)t
                                                               −                                                         :
                                                                           e t − e −t                   1
                                                                                                        2 − ix


Therefore, the eigenvalue equation det I λ − H(Γ; t ) = 0 is equivalent to the solubility
of the linear system
                    t                        
          1     2          iγu       −i γ  u
     λ      + γ zγ =     e         e          zγ  du −
          4           −t                      γ

                                       1                1                                             
                                   e ( 2 +i γ )t − e −( 2 +i γ )t  ( 12 −i γ  )t     −( 12 −i γ  )t
                               −                                     e             − e                   zγ  −
                                   ( 12 + i γ )(e t − e −t ) γ 
                                                 1                1                                             
                                             e ( 2 −i γ )t − e −( 2 −i γ )t  ( 12 +i γ  )t     −( 12 +i γ  )t
                                         −                                     e             − e                   zγ  :
                                              ( 12 − i γ )(e t − e −t ) γ 

      We write for simplicity
                                                         −i γ u
(8.5)                                        Z (u) =     e       zγ
                                                         γ

and then the above linear system takes the form
                       t                   1                1
           1                              e ( 2 +i γ )t − e −( 2 +i γ )t t=2
       λ     + γ 2 zγ = e i γ u Z (u) du − 1                             e Z (t ) − e −t=2 Z (−t ) −
           4              −t              ( 2 + i γ )(e t − e −t )
(8.6)
                                               1                 1
                                           e ( 2 −i γ )t − e −( 2 −i γ )t t=2
                                         − 1                              e Z (−t ) − e −t=2 Z (t ) :
                                            ( 2 − i γ )(e t − e −t )
    We note that since Γ = Γ we have
                 iγu          iγu                    
(8.7)               e (zγ ) =    e (zγ ) =    e −i γ u zγ = Z (u):
                     γ ∈Γ                γ ∈Γ                      γ ∈Γ

By differentiating (8.7), we see that
                                 
(8.8)                                 i γ e i γ u (zγ ) = Z  (u)
                                          γ ∈Γ

and
                                        
(8.9)                                          γ 2 e i γ u (zγ ) = ∆Z (u):
                                        γ ∈Γ


      We multiply both sides of (8.6) by ( 14 + γ 2 ) (zγ ) and sum over γ . In view of
remarks on weil’s quadratic functional in the theory of prime numbers, I                                     211


equations (8.7), (8.8) and (8.9) we find
                 1        2
                              2             t       
                                                        1
                                                             
             λ          +γ       zγ (zγ ) =     Z (u)     + ∆ Z (u) du +
                      4                      −t         4
                  γ ∈Γ


                                                   + Z  (t ) Z (t ) − Z  (−t ) Z (−t )−
(8.10)
                                                        e t + e −t                       
                                                  −        t    −t  |Z (t )|2 + |Z (−t )|2 +
                                                       2(e − e )
                                                                     1                                        
                                                            +    t        −t    Z (t ) Z (−t ) − Z (t ) Z (−t ) :
                                                                e −e
   Integration by parts shows that
          t                                                           t
                                                                            2
             Z (u) ∆Z (u) du = −Z  (t ) Z (t ) + Z  (−t ) Z (−t ) +     Z (u) du
             −t                                                                      −t

and (8.10) becomes
                1       2           
                                      1 t       2
                                                          t
                                                               2
             λ       + γ 2 zγ (zγ ) =        Z (u) du +     Z (u) du −
                   4                  4 −t                −t
                   γ ∈Γ


(8.11)                                                  e t + e −t        2         2 
                                                   −       t    −t     Z (t ) + Z (−t ) +
                                                       2(e − e )
                                                                     1                                        
                                                            +   t         −t    Z (t ) Z (−t ) − Z (t ) Z (−t ) :
                                                                e −e
   A further simplification is achieved if we decompose Z (u) into its even and odd
parts, namely
                                       1             
                              Z ± (u) = Z (u) ± Z (−u) :
                                       2
Then we verify that
          e t + e −t        2         2    1                                   
             t    −t     Z (t ) + Z (−t ) − t     −t Z (t ) Z (−t ) − Z (t ) Z (−t ) =
         2(e − e )                              e −e

                                                    e t=2 − e −t=2  + 2 e t=2 + e −t=2  − 2
                                               =                     Z (t ) + t=2            Z (t ) :
                                                    e t=2 + e −t=2            e − e −t=2
Also, for any function g we have
                    t                t                  t
                       |g (u)|2 du =     |g + (u)|2 du +     |g − (u)|2 du:
                          −t              −t                         −t

   Finally, since
                                            2
                                    1
                                      + γ2        zγ (zγ ) = wγ (wγ ) ;
                                    4
212                                                                                         e. bombieri

we can transform (8.11) into
                                   1  t  ± 2                t
                                                                       ±  2
                                                                                   
             λ        wγ (wγ ) =                       
                                               Z (u) du +                      
                                                                        (Z ) (u) du −
                                        4 −t                      −t
                 γ ∈Γ               ±
(8.12)
                                               e t=2 − e −t=2  + 2 e t=2 + e −t=2  − 2
                                           − t=2                Z (t ) − t=2            Z (t ) :
                                              e + e −t=2                   e − e −t=2
    Now we prove that
                                                                 t
            e t=2 ∓ e −t=2  ± 2 1 t  ± 2                      ±  2
                                                                     (Z ) (u) du
(8.13)                       Z (t )   ≤       Z     (u)   du  +
            e t=2 ± e −t=2              4 −t                      −t

for any continuously differentiable even or odd function Z ± (u). Furthermore, equality
holds if and only if
                                            e u=2 ± e −u=2
                                       Z ± (u) = Z ± (t )   :
                                             e t=2 ± e −t=2
    We verify this statement as follows. By direct calculation, equality holds in (8.13)
if Z ± (u) = e u=2 ± e −u=2 . Let
                                                                e u=2 ± e −u=2
                                F ± (u) = Z ± (u) − Z ± (t )                   :
                                                                e t=2 ± e −t=2
Then using the fact that e u=2 ± e −u=2 yields equality in (8.13), and that F ± (u) is even
or odd we verify by integration by parts the equation
                          t
     1 t  ± 2               ±  2
                               (F ) (u) du =
           F (u) du +
     4 −t                   −t

                                 t                       t
                                        ± 2                   ±  2         t=2  −t=2         
                        =
                            1          Z (u) du +            (Z ) (u) du − e ∓ e      Z ± (t )2 :
                            4     −t                        −t
                                                                                t=2
                                                                               e ±e  −t=2


In conclusion, we have

                                                t                    t
                                          1                                 2
(8.14)              λ          wγ (wγ ) =             F (u)2 du +         F (u) du
                                          4      −t                    −t
                        γ ∈Γ

with

(8.15)                                 F (u) = Z (u) − Ae u=2 − Be −u=2
and A, B determined by the condition F (t ) = F (−t ) = 0.
    This proves our statement.
    The proof of Lemma 10 follows from (8.14). Suppose first that every γ has multi-
plicity 1. If we had λ = 0 then (8.14) and (8.15) show that F (u) vanishes identically
in the interval (−t; t ). On the other hand, the functions e u=2 , e −u=2 and e −i γ u are
linearly independent over any finite interval because here Γ is a finite set; hence, we
would have zγ = 0 for every γ , a contradiction.
remarks on weil’s quadratic functional in the theory of prime numbers, I              213


   The general case can be treated much in the same way. It is easy to show that the
                                                              
spectrum of H(Γ; t ) consists of the point 0 with multiplicity  [m(γ ) − 1] together
with the spectrum of the matrix

                                   H (γj ; γk ; t )m(γk )

where now {γj } is the set of all distinct γ ∈ Γ. The same argument as before goes
through, provided we count each zγ with its multiplicity m(γ ).
   The last conclusion of Lemma 10 is obvious from (8.14), because if every γ is real
then (8.14) becomes
                                                   t
                                 1 t       2           2
                  λ     |wγ |2 =        F (u) du +     F (u) du:
                                 4 −t                −t
                      γ ∈Γ



    Theorem 8. The number of negative eigenvalues of the matrix H(Γ; t ) equals the number
of distinct complex conjugate pairs (γ ; γ ) in Γ.

    Proof. Suppose first that all elements of Γ are distinct and consider a continuous
deformation of Γ into a new set Γ    , such that during the deformation the elements
of the set remain distinct and the invariance of the set by complex conjugation and
multiplication by −1 remains preserved. By Lemma 10 all eigenvalues remain real and
not 0, while moving continuously during the deformation; in particular, the number of
negative and positive eigenvalues remains constant.
    Thus Theorem 8 for a multiset Γ with distinct elements will follow, once it has been
verified for any other set  Γ with distinct elements, invariant by complex conjugation
and multiplication by −1, with the same number of complex conjugate pairs of elements
as Γ. The general case then is an immediate consequence of Lemma 10.
    Now we verify Theorem 8, for a multiset Γ with distinct elements, by induction
on the number of complex conjugate pairs. Note that, according to our preceding
discussion, to complete the induction step it suffices to verify it adding to Γ some new
element γ0 and its transforms by complex conjugation and by multiplication by −1.
This element γ0 is otherwise at our disposal.
    By Lemma 10, Theorem 8 is true if every γ is real. We examine separately the case
in which we increase our multiset by {γ0 ; γ 0 ; −γ0 ; −γ 0 } with γ0 complex and not
purely imaginary, or by {γ0 ; γ 0 } with γ0 purely imaginary.
    Suppose Theorem 8 holds for Γ and let us show that it remains true for the new
       
set Γ {γ0 ; γ 0 ; −γ0 ; −γ 0 } where γ0 = a + ib, provided a and b are large enough.
    In what follows, we keep Γ fixed and consider γ0 as a variable quantity, so our
estimates will be uniform only with respect to γ0 . In the end, we let
                                 a=b → 0;      b →+ ∞:

   A simple asymptotic calculation shows that with the above choice of γ0 the matrix
214                                                                                          e. bombieri
                
 H (γ ; γ  ; t ) for γ ; γ  ∈ {γ0 ; γ 0 ; −γ0 ; −γ 0 } is (9)
                                                                               
                          o(B)           B + o(B) o(B)                 o(B)
                         B + o(B) o(B)               o(B)             o(B)     
                                                                               
                         o(B)           o(B)         o(B)             B + o(B) 
                          o(B)           o(B)         B + o(B)         o(B)
where for simplicity we have written
                                                    sinh(2tb)
                                              B=                :
                                                     2tb |γ0 |2
      The eigenvalue equation is obtained by equating to 0 the determinant of
                                                                                                    
  o(B) − x        B + o(B)             o(B)             o(B)              H(γ0 ; γ  ; t )
 B + o(B)
                 o(B) −x              o(B)             o(B)              H(γ 0 ; γ  ; t )         
                                                                                                     
 o(B)            o(B)                 o(B) − x         B + o(B)          H(−γ0 ; γ  ; t )         :
                                                                                                    
 o(B)            o(B)                 B + o(B)         o(B) − x                       
                                                                          H (−γ 0 ; γ ; t )          
  H (γ ; γ0 ; t ) H(γ ; γ 0 ; t )                                                  
                                       H (γ ; −γ0 ; t ) H (γ ; −γ 0 ; t ) H (γ ; γ ; t ) − δγ ;γ  x
We know already that this matrix has only real eigenvalues.
   Let D(x) denote this determinant. We expand D(x) according to a block Laplace
expansion using the first four rows of the matrix. The entries of the matrix (save for
the term x) are of order
                 O(B)            for the entries in the 4 × 4 upper-left corner ;
(8.16)           o(B)            for the other entries in the first four rows or columns ;
                 O(1)            for all other entries.
    If M × M is the size of our matrix, we obtain using (8.16) the approximation
                                                                              
(8.17)      D(x) = (x 2 − B 2 )2 det Ix − H(Γ; t ) + o (B + |x |)4 (1 + |x |)M −4 :
      Suppose first that x = O(1). Then (8.17) shows that
                                           
                            B −4 D(x) = det Ix − H(Γ; t )] + o(1):
It follows that D(x) has, for a=b → 0, b →+ ∞, M − 4 real roots arbitrarily close to
the eigenvalues of H(Γ; t ), and in particular of the same sign.
    If instead x is of order B we have
                                                
                                det Ix − H(Γ; t ) ∼ x M −4
and setting x = By we get
                                  B −4 x −M +4 D(x) = (y 2 − 1)2 + o(1):
Therefore, D(x) has, for a=b → 0, b →+ ∞, two real roots asymptotic to −B and
two real roots asymptotic to B. Since we have accounted for all roots of D(x), the set

      (9) We write o(B) to indicate suitable entries of that order of magnitude.
remarks on weil’s quadratic functional in the theory of prime numbers, I                215

  
Γ {γ0 ; γ 0 ; −γ0 ; −γ 0 } again satisfies the conclusion of Theorem 8, completing the
induction step.
   It remains for consideration the case in which γ0 = ib is purely imaginary. This
time the determinant equation is
                                                                                  
                       o(B) − x        B + o(B)         H (γ0 ; γ  ; t )
        D(x) = det  B + o(B)          o(B) − x         H (γ 0 ; γ  ; t )          = 0:
                                                                 
                       H (γ ; γ0 ; t ) H (γ ; γ 0 ; t ) H (γ ; γ ; t ) − δγ ;γ  x
The same argument as before shows that, as a=b → 0, b →+ ∞, besides the roots
asymptotic to the eigenvalues of H(Γ; t ) we acquire one real root asymptotic to B and
another real root asymptotic to −B, completing the proof of Theorem 8.

           9. The second eigenvalue problem: even and odd eigenfunctions
    We will need the analogue of Theorem 8 for the second eigenvalue problem. Our
problem is
                                    i(γ −γ  )u 
(9.1)                     λ zγ =        e         du zγ  :
                                       γ          E

We shall assume that E is a finite union of bounded closed intervals. We say that E
is symmetric if
(9.2)                                           E = −E:
The associated eigenfunction is
                                        
(9.3)                        F (u) =            zγ e −i γ u     for u ∈ E
                                            γ

and F (u) = 0 for u ∈
                    = E.
   If E is symmetric then
                                                      
                                                                
                                  e i(γ −γ )u du =            e i(γ −γ )u du;
                              E                          E

and since Γ = −Γ we see that if {zγ } is a solution of the linear system (9.1) then {z−γ }
is another solution, for the same eigenvalue. Therefore, by considering { 12 (zγ ± z−γ )}
we see that we have a basis of solutions which are either even or odd, namely such that
(9.4)                              zγ = ±z−γ            for γ ∈ Γ:
Accordingly, the corresponding eigenfunction is even or odd in the usual sense
(9.5)                                  F (−u) = ±F (u):
Hence
   Lemma 11. If E is symmetric there is a basis of solutions of (9:1) consisting of even or
odd eigenvectors {zγ } satisfying zγ = ±z−γ . The associated eigenfunctions satisfy F (−u) =
= ±F (u).
216                                                                                                          e. bombieri

    We can separate the space of solutions of (9.1) into its even and odd parts as follows.
Let us define
                                                    
(9.6)                   KE± (x; y) = e i(x −y)u du ± e i(x +y)u du
                                                  E                       E

and define
                  %                     &'%                                &
(9.7)         Γ0 = γ : γ ∈ Γ; (γ ) > 0     γ : γ ∈ Γ; (γ ) = 0; (γ ) > 0 :
Then Γ − {0} = Γ0 ∪ −Γ0 .
         = Γ the linear system (9.1) splits into two new linear systems
    If 0 ∈
                                 ±
(9.8)                    λ zγ =      KE (γ ; γ  )zγ  ; γ ∈ Γ0 ;
                                       γ  ∈ Γ0

where if 0 ∈ Γ one should add a term 12 KE+ (γ ; 0)z0 to the right-hand side of the
equation in the + case.
    We have the following analogue and strengthening of Lemma 10 for the second
eigenvalue problem.
                                                                           = Γ. Then the
   Lemma 12. Suppose that E is symmetric and suppose for simplicity that 0 ∈
matrix KE (Γ) is equivalent to the direct sum of the matrices

                                 KE± (Γ) = KE± (γ ; γ  )                              :
                                                                        γ ;γ  ∈ Γ 0

    A matrix KE± (Γ) admits the eigenvalue 0 if and only if there is γ ∈ Γ0 with multiplicity
                                                                                
greater than 1, in which case the multiplicity of 0 as an eigenvalue is exactly    [m(γ ) − 1],
        
where      ranges over all distinct γ ∈ Γ0 .
    Moreover if every γ is real all eigenvalues of KE± (Γ) are non-negative.
  Remark. If 0 ∈ Γ the result continues to hold, modifying the definition of KE+ (Γ)
when γ or γ  equal 0.
      Proof. Consider first the even case. Let {zγ } be an even solution of (9.1), hence
                                                      z−γ = zγ :
For γ ∈ Γ0 we have
                              
                     λ zγ =            K E (γ ; γ  ) z γ  =
                              γ  ∈Γ

                                                                    
                          =                K E (γ ; γ  ) z γ  +                 KE (γ ; γ  ) zγ  =
                              γ  ∈Γ   0                            γ  ∈−Γ   0

                                                                      
                          =    KE (γ ; γ  ) zγ  + KE (γ ; −γ  ) z−γ  =
                              γ  ∈ Γ0

                                                                           +
                          =       KE (γ ; γ  ) zγ  + KE (γ ; −γ  ) zγ  =  KE (γ ; γ  ) zγ  :
                              γ  ∈ Γ0                                                             γ  ∈Γ0
remarks on weil’s quadratic functional in the theory of prime numbers, I                                                   217


   Exactly the same argument works in the odd case. This proves the first part of the
lemma.
   The proof of the second part is equally easy. For γ ∈ Γ we have
                                                     
                                                     iγu
                    λ    zγ (zγ ) =          (zγ ) e       F (u) du =
                         γ ∈Γ                   E       γ

                                                                             
                                                                       −i γ u
                                        =                       zγ e                F (u) du =
                                                E       γ

                                                                                                          
                                                                                                                       
                                                    =                      zγ e     −i γ u
                                                                                                 F (u) du =       F (u)2 du :
                                                            E          γ                                      E

If {zγ } is either even or odd we have
                                               
                                   zγ (zγ ) = 2   zγ (zγ ):
                                 γ ∈Γ                       γ ∈Γ 0

Hence                                                       
                                                                         
                                2λ            zγ (zγ ) =             F (u)2 du
                                     γ ∈ Γ0                      E

and we conclude as in Lemma 10.
    Theorem 9. Let E be a finite union of bounded closed intervals. Then the number of
negative eigenvalues of KE (Γ) equals the number of distinct complex conjugate pairs {γ ; γ }
in Γ.
    Suppose also that E is symmetric. Then the number of negative eigenvalues of KE+ (Γ) equals
the number of distinct complex conjugate pairs (γ ; γ ) with (γ ) > 0, and the number of
negative eigenvalues of KE− (Γ) equals the number of distinct complex conjugate pairs (γ ; γ )
with (γ ) ≥ 0.
    Proof. The proof follows the proof of Theorem 8, with some minor modifications.
It suffices to prove it in the case in which all γ ’s are distinct, proceeding by induction
on the number of complex conjugate pairs in Γ, or Γ0 if we are dealing with the
symmetric case. We give some details only in the symmetric case.
    Suppose Theorem 9 holds for Γ0 and let us show that it continues to hold for the
             
new set Γ0 {γ0 ; γ 0 }, for some γ0 = a + ib with a ≥ 0 and b > 0.
    Suppose first that a > 0. As in the proof of Theorem 8, it suffices to study the
zeros of the determinant
                       ±                                                                 
                        KE (γ0 ; γ0 ) − x KE± (γ0 ; γ 0 )      KE± (γ0 ; γ  )
                       ±                                                                 
        D ± (x) = det 
                       KE (γ 0 ; γ0 )    KE± (γ 0 ; γ 0 ) − x KE± (γ 0 ; γ  )           :
                                                                                          
                        KE± (γ ; γ0 )     KE± (γ ; γ 0 )       KE± (γ ; γ  ) − δγ ;γ  x
We already know that the zeros of D ± (x) are all real.
218                                                                                      e. bombieri

      Let t be the largest element in E . We let b →+ ∞ and a=b → 0. Then
                                        
                                                           
                       KE± (γ0 ; γ0 ) =     1 ± e (2ia −2b)u du = o(e 2bt =b)
                                              E
                                              
                                             −2bu                 e 2bt
                          KE± (γ0 ; γ 0 ) =   e      ± e 2iau du ∼
                                          E                          2b
                                                                   2bt
                                             2bu                e
                      KE± (γ 0 ; γ0 ) =       e ± e 2iau du ∼
                                          E                        2b
                                         
                                                             
                      KE± (γ 0 ; γ 0 ) =      1 ± e (2ia +2b)u du = o(e 2bt =b):
                                              E

                    2bt
If we write B = e     =(2b) and M is the cardinality of Γ0 we see that
                                                                                    
             D ± (x) = (x 2 − B 2 ) det[Ix − KE± (Γ)] + o (B + |x |)2 (1 + |x |)M −2

and we argue as in the proof of Theorem 8.
   If however γ0 = ib is purely imaginary we have
                                
                                             
                  ±
                 K (γ0 ; γ0 ) =     1 ± e −2bu du ∼ ±e 2bt =(2b) = ±B:
                                          E

The same argument shows that D + (x) has a real root asymptotic to B while D − (x) has a
real root asymptotic to −B; all other roots are asymptotic to the roots of det[Ix −KE± (Γ)].
This completes the proof.

                                     10. The passage to the limit

    In this section we pass to the limit from the finite sets ΓN to infinite sets Γ. The
simplest situation is when the set of complex pairs of ΓN is finite.
    Thus let Γ be an infinite multiset of complex numbers, satisfying as always the
                         
convergenge condition       1=(1 + |γ |)1+ε <+ ∞ for every ε > 0 and the symmetry
conditions Γ = Γ, Γ = −Γ. By ΓN we denote the truncations of Γ at |γ | ≤ N .
    Suppose that Γ has finitely many, but at least one, complex pairs of elements and
let N be so large that ΓN contains all complex pairs of Γ. Then H(ΓN ; t ) has at least
one negative eigenvalue λN < 0, which we may choose to be the largest in absolute
value.
    Let {wγ ;N } be a solution of the corresponding eigenvalue equation
                                        
                           λN wγ ;N =        H (γ ; γ  ; t ) wγ  ;N
                                                  γ  ∈Γ N


normalized so that
                                              
(10.1)                                               |wγ ;N |2 = 1:
                                              γ ∈Γ
remarks on weil’s quadratic functional in the theory of prime numbers, I                                 219


   Then equation (8.14) shows that
                                                        t
                                     1 t
(10.2)       λN     wγ ;N (wγ ;N ) =      |FN (u)|2 du +     |FN (u)|2 du
                                     4 −t                 −t
                    γ ∈Γ

where
                                      
(10.3)                     FN (u) =            zγ ;N e −i γ u − Ae u=2 − Be −u=2 ;
                                      γ ∈Γ N


zγ ;N = wγ ;N =( 14 + γ 2 ) and FN (t ) = FN (−t ) = 0.
    Now we note that, since λN < 0, we have
                                                                   
               λN        wγ ;N (wγ ;N ) = λN      |wγ ;N |2 = −|λN |   |wγ ;N |2 :
                    γ ∈R                           γ ∈R                              γ ∈R

Therefore, after division by |λN |, (10.2) becomes
                                               t                t            
                                            1   1                      2
   −     wγ ;N (wγ ;N ) =     |wγ ;N |2 +            F (u)2 du +     F (u) du
                                          |λN | 4 −t N              −t
                                                                         N
       =R
      γ∈                     γ ∈R

and in particular
                                                               
                               −          wγ ;N (wγ ;N ) ≥             |wγ ;N |2 :
                                     =R
                                    γ∈                          γ ∈R

Hence by Cauchy’s inequality we infer
                                           
                                |wγ ;N |2 ≥   |wγ ;N |2
                                      =R
                                     γ∈                   γ ∈R

and conclude with the lower bound
                                           
                            2   |wγ ;N |2 ≥   |wγ ;N |2 :
                                       =R
                                      γ∈                     γ ∈Γ

This inequality, in conjunction with (10.1), shows that

   Lemma 13. If λN < 0 we have
                                          1 
                                            ≤ |wγ ;N |2 ≤ 1
                                          2
                                                 =R
                                                γ∈

and |wγ ;N | ≤ 1 for every γ ∈ ΓN .
                                                                                
    Now we recall that zγ ;N = wγ ;N =( 14 + γ 2 ) and that                           1=(1 + |γ |)2 <+ ∞. It
follows that the sum
                                          
                              ZN (u) =        zγ ;N e −i γ u
                                                    γ ∈Γ N

is uniformly bounded and majorized by an absolutely convergent series. In fact, more
than this is true, because ( 14 + γ 2 ) zγ ;N  is uniformly bounded by 1; therefore, there is
220                                                                                           e. bombieri

a subsequence of N → ∞ such that each zγ ;N converges to a limit zγ , again satisfying
 1           
( + γ 2 ) z  ≤ 1. It follows that Z (u) converges pointwise and uniformly to the limit
  4         γ                        N
                                             −i γ u
                                    Z (u) =    zγ e
                                                     γ ∈Γ

in compact subsets of R. In particular, Z (t ) and Z (−t ) are bounded and the functions
FN (u) converge pointwise and uniformly to a limit
                                  −i γ u
                         F (u) =     zγ e      − Ae u=2 − Be −u=2
                                     γ ∈Γ

with A and B determined by the condition F (t ) = F (−t ) = 0.
    Now note that the eigenvalue λN is uniformly bounded from below.
    This is seen as follows. ByLemma 13, if there are J complex γ ∈ Γ then for one
of them we have |wγ ;N | ≥ 1= 2J . Thus the equation
                                      
                           λN wγ ;N =    H (γ ; γ  ; t ) wγ  ;N
                                            γ  ∈Γ N

shows that
                                                                                     1=2
                                                                               
                       |λN | ≤    2J max                       H (γ ; γ  ; t )  2
                                                     γ  ∈Γ

where the maximum runs over the finitely many complex γ ∈ Γ, proving our claim.
   The basic equation (10.2) also shows that
                                  t                    
              1 t        2            2
(10.4)               FN (u) du +      F (u) du ≤ |λ |
                                         N            N    |wγ ;N |2 = |λN |:
              4 −t                  −t                                          γ ∈Γ N

By semicontinuity, from (10.4) we deduce
                                    t
               1 t
(10.5)                |F (u)|2 du +     |F  (u)|2 du ≤ lim inf |λN | <+ ∞:
               4 −t                  −t
                                                         N →∞

   Suppose that F (u) is not identically 0 in (−t; t ). Then λ = lim infN →∞ λN < 0
and, keeping in mind the rapid decay of H (γ ; γ  ; t ) with γ  , we see that
                                     
(10.6)                       λ wγ =      H (γ ; γ  ; t ) wγ 
                                            γ  ∈Γ

with wγ = ( 14 + γ 2 ) zγ .
   We have verified already that a solution of (10.6) yields a solution of the eigenvalue
problem
                                  λDf (x) = L[f ](x)
for x ∈ (M −1 ; M ), M = e t , with f ∈ W0 . Indeed, F (u) = e u=2 f (e u ). Moreover, for
any α < 1 we have Df ∈ C 1;α in the interval (M −1 ; M ) and Df is not identically 0
because F (u) is not identically 0 by hypothesis. It follows that
                                             ∗                      2
(10.7)                           T [f ∗ f ] = λ Df                      < 0:
remarks on weil’s quadratic functional in the theory of prime numbers, I                            221


    On the other hand, since this holds for any t > 0, we can choose t < 12 log 2, hence
       √                          ∗
M < 2. It follows that f ∗ f has compact support in (1=2; 2) and in particular in
Weil’s Explicit Formula the terms involving Λ(n) are absent.
    It is possible (10) to prove directly the positivity of Weil’s Explicit Formula in this
restricted case where the support of f is [M0−1 ; M0 ], with M0 > 1 an explicitly com-
putable constant. This contradicts (10.7). Therefore, we conclude with

                                                                                      = R,
   Theorem 10. Suppose ζ (s) has only finitely many non-trivial zeros 12 + i γ with γ ∈
and at least one such zero.
   Then for t > 0 there are complex coefficients wγ and λ ≤ 0 such that
                        
                            |wγ |2 = 1;
                        γ ∈Γ
                                                                  1
                               |wγ |2 ≥ −           wγ (wγ ) ≥        ;
                                                                    2
                         =R
                        γ∈                     =R
                                              γ∈
                                  
                       λ wγ =              H (γ ; γ  ; t ) wγ      for every γ ∈ Γ;
                                  γ  ∈Γ

and with the following property. Let
                                                           wγ
                                      Z (u) =             1    2
                                                                     e −i γ u
                                                      γ   4 +γ

and
                           e t=2 Z (t ) − e −t=2 Z (−t ) u=2 e t=2 Z (−t ) − e −t=2 Z (t ) −u=2
         F (u) = Z (u) −                                e −                               e     :
                                      e t − e −t                      e t − e −t
   Now define
                                     f (x) = φ(x) x −1=2 F (log x)
where φ(x) is the characteristic function of [e −t ; e t ]. Then either
                                                  ∗                  2
                                     T [f ∗ f ] = λ Df                   <0
or f (x) is identically 0. In addition, if 0 < t < t0 where t0 is a suitable explicitly computable
constant, the function f (x) must be identically 0.


                 11. The second eigenvalue problem: linear independence
    The considerations of the preceding two sections can also be done in the setting in
                                                       ∗
which we minimize the quadratic functional T [f ∗ f ]= f 2 in L2 (E ) rather than with
respect to the norm Df = 1.
    Let E ⊂ R be a finite union of disjoint intervals and consider the eigenvalue problem
(11.1)                             λF (u) = L[F ](u)               for u ∈ E:

   (10) We defer the proof to §12, Theorem 12.
222                                                                                                                 e. bombieri

      As in Section 8, we get the equation
                                          −i γ u
                                 F (u) =   zγ e   ΦE (u)
                                                   γ ∈Γ

where ΦE is the characteristic function of E , and the eigenvalue equation
                                           
                                                           
(11.2)                      λ zγ =      zγ  e i γ u e −i γ u du:
                                              γ  ∈Γ        E

Again, this yields
                                                           
                                                                         
(11.3)                             λ          zγ (zγ ) =            F (u)2 du:
                                       γ ∈Γ                     E

Thus λ = 0 if Γ is a finite set. Suppose λ < 0 and normalize {zγ } to have ‘2 -norm
equal to 1. Then, exactly as in the preceding section, we have
                                               1
(11.4)                                  |zγ |2 ≥ :
                                                2
                                                 =R
                                                γ∈

    We apply this to the finite set ΓN and the negative eigenvalue λN and let N → ∞,
keeping λN < 0 largest in absolute value. Let {zγ ;N } be the corresponding eigenvector
as in (11.2). We may assume by going to a subsequence that
                                               zγ = lim zγ ;N
                                                        N →∞
                                                                                                                    
exists for every γ , and now (11.4) continues to hold. The limit F (u) =                                                zγ e −i γ u
exists as an L2 -function and is a weak limit of the sequence
                                          
                                 FN (u) =      zγ ;N e −i γ u :
                                                       γ ∈ ΓN
                                          
Note that (11.3) implies FN ≤                  −λN , hence
(11.5)                                            F 2 ≤ −λ;
with λ = lim λN . Hence if λ = 0 we must have F (u) = 0 identically on E .
   Next, we note that (11.2) continues to remain true in the limit. We have, for
K ≥ 1 and N > (K + 1)|γ |, the bound
                                            2                                    2
                             i(γ −γ  )u
                                                                     i(γ −γ  )u
                                                                                       
                  zγ  ;N   e                
                                           du  ≤                  
                                                      |zγ  ;N | ·  e             du  ≤
     
         (K +1)|γ |<|γ  |≤N   E                                    K |γ |<|γ  −γ |                      E
                                                                                                                           2
                                                                                                                           
                                                         ≤ {zγ ;N } 2‘2 ·                                  e i(γ −γ  )u du  
                                                                                                                            
                                                                                       |γ  −γ |>K |γ |       E
                                                                                       1
                                                                                                  
                                                                |γ  −γ |>K |γ |
                                                                                 (1 + |γ − γ  |)2
                                                                   log(n + 1 + |γ |)              log(K |γ | + 2)
                                                                                                                                :
                                                                n>K |γ |
                                                                                 (n + 1)2                         K |γ | + 2
remarks on weil’s quadratic functional in the theory of prime numbers, I                                                     223


Hence if N > (K + 1)|γ | we have
                                     
                                                
          0 = λN zγ ;N −      zγ  ;N   e i(γ −γ )u du =
                             |γ  |≤N               E
                                                          
                                                                            
              = λN zγ ;N −                       zγ  ;N         e i(γ −γ )u du +
                             |γ  |≤(K +1)|γ |               E
                                                                                                  
                                                                                   i(γ −γ  )u
                                                                                                      
                         + O                                 zγ  ;N          e                 du  =
                                       (K +1)|γ |<|γ  |≤N                  E

                                                                                                                      1=2 
                                                                     i(γ −γ  )u                       log(K |γ | + 2)
              = λN zγ ;N −                       zγ  ;N         e                 du + O                                         :
                                                             E                                           K |γ | + 2
                             |γ  |≤(K +1)|γ |


This estimate is uniform in N and the sum over |γ  | ≤ (K + 1)|γ | is a finite sum
indexed independently of N . Thus for fixed γ and K we can pass to the limit as
N → ∞, obtaining
                                                                    1=2 
                                                      log(K |γ | + 2)
         0 = λ zγ −          zγ  e i(γ −γ )u du + O                          :
                                 E                      K |γ | + 2
                     |γ |≤(K +1)|γ |


Now we let K → ∞ and conclude that
                                i γ u −i γ  u
                       λ zγ =      zγ  e e      du
                                               γ            E

for every γ , as asserted.
    This equation can be rewritten as
                                
                                   e i γ u F (u) du = λzγ ;
                                          E

hence noting that |(γ )| ≤ 12        we infer
                              
                                                   
                                      e |u|=2 F (u) du ≥ |λ| max |zγ | :
                                E

Therefore, by (11.4) we find
                           
                                  
(11.6)                       F (u) du ≥ (2J )−1 e −m(E )=2 |λ|
                                  E

where J is the number of complex zeros of ζ (s) off the critical line and m(E ) = maxE |x |.
In particular, in view of (11.5) we have shown

   Lemma 14. We have F (u) = 0 identically in E if and only if λ = lim inf λN = 0.

   This gives the analogue of Theorem 8 in this setting.

  Theorem 11. Suppose that ζ (s) has only finitely many non-trivial zeros 12 + i γ with
 = R, and at least one such zero. Then for any finite union of intervals E there are complex
γ∈
224                                                                                       e. bombieri

coefficients zγ = zγ (E ) and λ = λ(E ) ≤ 0 such that
                                
                                   |zγ |2 = 1;
                                   γ ∈Γ
                                                                          1
                                          |zγ |2 ≥ −          zγ (zγ ) ≥
                                                                            2
                                    =R
                                   γ∈                   =R
                                                       γ∈

and with the following property. Let
                                                     
                                          F (u) =           zγ e −i γ u :
                                                     γ


Then F (u) is locally in L2 and
                                     
                            λ zγ =        e i γ u F (u) du      for every γ ∈ Γ:
                                      E

Moreover, if
                                     f (x) = φ(x) x −1=2 F (log x)
where φ(x) is the characteristic function of E = e E , we have either
                                                 ∗                 2
                                     T [f ∗ f ] = λ f                  <0
or λ = 0 and f (x) is identically 0.
      As before, we have
    Corollary. Let E ⊂ (0; ∞) be a finite union of bounded closed intervals and suppose that
        ∗
T [f ∗ f ] ≥ 0 for every smooth function f with compact support in E .
    Then either the Riemann Hypothesis is true, or ζ (s) has infinitely many zeros off the critical
line, or the functions x −ρ are linearly dependent over E , for a suitable sequence of coefficients
{cρ } ∈ ‘2 such that at least half of its ‘2 -mass is supported on the set of zeros ρ with (ρ) = 12 .

   One may ask if linear dependence relations occur at all. The following example
shows that they may occur for Dedekind zeta functions.
    An Example. Consider two Dirichlet L-functions L(s; χ) and L(s; χ ) for a same
modulus q > 1 and two distinct primitive characters with the same parity, hence
χ(−1) = χ (−1). Let also p0 ≥ 2 be the first prime for which χ(p0 ) = χ (p0 ).
    These two Dirichlet L-functions have the same conductor and Gamma factors in
their functional equations. Consequently, the Explicit Formula associated with these two
functions will have the same contribution from the «prime at infinity». Moreover, if we
evaluate the Explicit Formula for a function f (x) with compact support in (1=p0 ; p0 ),
the contribution arising from the primes p < p0 is the same in both cases, and it is 0
for the primes p ≥ p0 because of the condition imposed on the support of f (x). By
taking the difference, we obtain the relation
                                           
                                   f(ρ) −     f(ρ ) = 0
                                      ρ                ρ
remarks on weil’s quadratic functional in the theory of prime numbers, I                         225


where ρ runs over the non-trivial zeros of L(s; χ) and ρ runs over those of L(s; χ ).
   Now fix f0 (x) with compact support in (e −ε ; e ε ) and let f (x) = f0 (ax) with e ε =p0 <
< a < e −ε p0 . Then f(s) = a −s f0 (s) and we get the relation
                                                              
                                f0 (ρ) a −ρ −    f0 (ρ ) a −ρ = 0
                               ρ                 ρ
           ε       −ε
for a ∈ (e =p0 ; e p0 ).
    As pointed out by J. Bourgain, the existence of linear relations over intervals of
arbitrary length also follows from the fact that the gap between consecutive γ ’s tends
to 0 as γ → ∞. In any case, it should be noted that the coefficients of the relations
in question are obtained as limits of eigenvectors of reasonably well-behaved matrices.
Computer experiments (11) suggest that if a negative eigenvalue λN for the problem
above approaches 0 as N → ∞, then it does so at an exponential or nearly exponential
rate and also the corresponding eigenvector seems to converge rapidly to a limit. Thus
a deeper study of the eigenvectors associated to finite approximations may shed some
light on this question of linear independence.
    We suggest that splitting the eigenvalue problem into its even and odd components
and complexifying it with the help of Cramér’s function
                                              iγz
                                    V (z) =      e
                                                 (γ )>0

may bring the problem to a form amenable to the extraordinarily powerful Riemann-
Hilbert techniques recently introduced in other contexts. An examination of these
possibilities may prove to be a valuable undertaking in studying the structure of the
limiting linear relations.

                                      12. Sets of positivity
    In this section we prove the positivity statement needed for the proof of the Corollary
to Theorem 8. Another proof can be found in Yoshida’s paper [5]. Let T [F ] be the
linear form
                                               
(12.1)                               T [F ] =     F( (γ )
                                                  γ ∈Γ

where F( denote the Fourier transform.
   Definition. We say that a closed subset E ⊂ R is a set of positivity for T if
        T [F (x) ∗ F (−x)] > 0 for every smooth function F with compact support in E
where                                          ∞
                              (F ∗ G )(x) =         F (u)G (x − u) du
                                               −∞



   (11) The experiments were done by adding a «fake zero» ρ0 off the critical line, together with its
symmetric images, to the first k non-trivial zeros of ζ (s), with k up to 320.
226                                                                                   e. bombieri

is the additive convolution, hence
                                                   ∞
(12.2)                      F (x) ∗ F (−x) =             F (x + u)F (u) du:
                                                    −∞

      In this setting, the Explicit Formula becomes
                   ∞                           ∞
                                                     Λ(n)                     
        T [F ] =       2 cosh(x=2) F (x) dx −        √ F (log n) + F (− log n) −
                   −∞                           n =1
                                                       n
                                                 ∞
                                                                                      dx
                        − (log 4π + γ ) F (0) −       e x=2 F (x) + F (−x) − 2F (0)            :
                                                    0                                 e − e −x
                                                                                       x

Moreover, the last two terms in this formula can be written as
                                        ∞
                                     1       Γ 1     v
                   −(log π )F (0) +            ( + i ) F( (v) dv;
                                    2π −∞ Γ 4         2
as one verifies using Theorem 2 and a change of variables.
      Theorem 12. If F (x) has compact support in an interval I of length |I | < log 2 we have
                                                                              
                                 (      (             1              1
                                                                         − O(1) F 2 :
                                                               +
        T [F (x) ∗ F (−x)] =     F (γ ) F (γ ) ≥ log      − log log
                                                     |I |           |I |
                             γ

    Proof. Since G (x) = F (x) ∗ F (−x) remains unchanged if we replace F (x) by a trans-
lation F (x + c), we may assume that F (x) is supported in the interval [−a=2; a=2] with
a < log 2. Then G (x) is supported in [−a; a] ⊂ (− log 2; log 2). Thus in the Explicit
Formula as above for T [G ], the contribution of the sum involving Λ(n) vanishes. It
follows that
                                     ∞
                          T [G ] =       2 cosh(x=2) G (x) dx −
                                     −∞
(12.3)                                                        ∞
                                                          1        Γ 1     v (
                                      − (log π ) G (0) +            ( +i ) G     (v) dv:
                                                         2π −∞ Γ 4          2
We have
(12.4)                           G (0) = F 2 ;          ( (v) = |F( (u)|2
                                                        G
and by Plancherel’s Formula
                                           ∞
                                      1
(12.5)                                          |F( (v)|2 dv = F 2 :
                                     2π   −∞

Now
               a                       
                                        
                  2 cosh(x=2) G (x) dx  ≤
              
                −a
(12.6)                                  a  ∞                    
                                                                  
                        ≤ 2 cosh(a=2)          F (x + u)F (u) du  dx ≤ 4a cosh(a=2) F 2 :
                                             
                                           −a      −∞

Since a < log 2 and
                                  Γ 1   v
                                   ( + i ) = log+ |v | + O(1);
                                  Γ 4    2
remarks on weil’s quadratic functional in the theory of prime numbers, I                         227


we infer from (12.3) to (12.6) that
                                                        ∞
                                                1
(12.7)              T [G ] ≥ −O(1) F 2 +                     (log+ |v |) |F( (v)|2 dv:
                                               2π      −∞

For any K > 1 we have
              ∞                                   
           1                                log K
                 (log+ |v |) |F( (v)|2 dv ≥             |F( (v)|2 dv =
          2π −∞                               2π |v |>K
                                                        
                                                  log K
(12.8)                        =   (log K ) F  2
                                                −               |F( (v)|2 dv ≥
                                                    2π |v |≤K
                                                                      ∞                    2
                                                                  1         sin(v=K )       
                                              2       2
                              ≥ (log K ) F − (4K log K )                              (
                                                                                       F (v) dv;
                                                                2π −∞           v
because | sin(v=K )| ≥ |v |=(2K ) for |v | ≤ K .
   The function (sin(v=K )=v)F( (v) is the Fourier Transform of the convolution
                                      1=K
                                             F (x − y) dy;
                                        −1=K

which is supported in the interval I = [− 2a − K1 ; 2a + K1 ].
   Therefore, by Plancherel’s Formula and Cauchy’s inequality we get
                ∞                 2       ∞   1=K                2
             1      sin(v=K )                                       
                              (    
                               F (v) dv =              F (x − y) dy  dx ≤
            2π −∞       v                   −∞
                                                 
                                                    −1=K
(12.9)                                         a=2           2
                                               ≤                 |F (y)| dy    dx ≤ (a + 2=K )a F 2 :
                                                   I     −a=2

By (12.7), (12.8), (12.9) we infer
                                                            
               T [G ] ≥ log K − O(1) − 4(a + 2=K )a K 2 log K · F 2 :
Theorem 12 follows by choosing
                                                    −1
                                       1           1
                                   K =     1 + log       :
                                       a           a


                              13. Some numerical experiments
    It is easy to follow numerically the behaviour of eigenvalues and eigenfunctions of
finite approximations, as the number of zeros increases.
    In this section we consider finite approximations to Problem 2, where the set of
points consists of the first N zeros of ζ (s) with positive imaginary parts and a fictitious
zero ρ0 off the critical line, together with their images by complex conjugation and
reflection about the point 1=2.
    The set E is the interval [−t; t ] with various ranges of t and the number N of
zeros ranges up to N = 160. The fictitious zero off the line has been arbitrarily set at
ρ0 = 0:52 + i3:14.
228                                                                                          e. bombieri

    In the symmetric case considered here there is a natural division of eigenvalues and
eigenfunctions into even and odd eigenfunctions. The numerical evidence gathered
here indicates in each case the existence of a critical value tc± > 0 such that the unique
negative eigenvalue λ±                           ±
                       N (t ) tends to 0 if t < tc , as N → ∞. The rate of convergenge
is quite fast, suggesting an exponential rate. For t > tc± this eigenvalue converges,
albeit less rapidly so, to a strictly negative value. The behaviour of the corresponding
eigenfunctions is markedly different for t < tc± and t > tc± , although even and odd
eigenfunctions behave rather similarly.
                                      2
                      1                               3               4                  5


       -2


       -4


       -6


       -8


      -10



                Fig. 1. – The negative even eigenvalue, ρ0 =:52+i3:14, N =10, 0<t<5:5.

                                      2
                      1                               3               4              5


      -2


      -4


      -6


      -8



      -10



                Fig. 2. – The negative odd eigenvalue, ρ0 =:52+i3:14, N =10, 0<t<0:55.
remarks on weil’s quadratic functional in the theory of prime numbers, I                             229

   The two plots together:
                                        2
                        1                               3               4               5


     -2


     -4


     -6


     -8



    -10



                  Fig. 3. – The two negative eigenvalues, ρ0 =:52+i3:14, N =10, 0<t<0:55.




   Normalized eigenfunctions, f             = 1, for t < tc+ .

                                                10


                                                 8


                                                 6


                                                 4


                                                 2




           -0.4                -0.2                                  0.2                0.4

                                                 -2


      Fig. 4. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =20, t =0:48.
230                                                                                          e. bombieri


                                             12.5

                                               10

                                              7.5

                                                5


                                              2.5



                              -0.2                                  0.2
           -0.4                                                                        0.4
                                             -2.5

                                               -5

      Fig. 5. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =40, t =0:48.




                                               15



                                               10



                                                5




                             -0.2                                   0.2                0.4
           -0.4

                                               -5



      Fig. 6. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =80, t =0:48.
remarks on weil’s quadratic functional in the theory of prime numbers, I                             231

   Normalized eigenfunctions, f           = 1, for t > tc+ .



                                             0.75


                                              0.5


                                             0.25



         -2                    -1                                    1                  2

                                             -0.25


                                              -0.5


                                            -0.75



       Fig. 7. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =40, t =2:3.




                                             0.75


                                              0.5


                                             0.25


          -2                   -1                                    1                  2

                                             -0.25


                                              -0.5


                                            -0.75



       Fig. 8. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =80, t =2:3.
232                                                                                            e. bombieri



                                              0.75


                                                0.5


                                               0.25



           -2                   -1                                     1                  2

                                              -0.25


                                               -0.5


                                              -0.75



        Fig. 9. – The eigenfunction for the negative even eigenvalue, ρ0 =:52+i3:14, N =160, t =2:3.


    Further computer experiments for t < tc+ and large N , both for the negative eigen-
value and a sequence of positive eigenvalues tending to 0 as N increases, point out to
a marked difference in behaviour for eigenvectors belonging to the negative eigenvalue
compared with eigenvectors belonging to positive eigenvalues.
    In the case t < tc± , normalized eigenfunctions f (x) with f = 1 converge weakly
to 0. The numerical experiments indicate that the L2 -mass of the function gets more
and more concentrated at the boundary of the interval (−t; t ), as N → ∞, and one
should study the asymptotic behaviour of eigenfunctions at the boundary, after the
appropriate rescaling. This should be of particular interest at the critical value t = tc± ,
which represent the transition from weak convergence to 0 to strong convergence in L2 .

                                         Acknowledgements
    The author wishes to thank here Tom Spencer, Percy Deift, Henryk Iwaniec and Jean Bourgain for
several interesting conversations. Theorem 8 was first discovered experimentally with the help of the
Mathematica computer package.



                                               References
[1] E. Bombieri - J.C. Lagarias, Complements to Li’s criterion for the Riemann Hypothesis. J. Number Theory,
    77, 1999, 274-287.
[2] A.P. Guinand, Summation Formulae and Self-reciprocal Functions (III). Quarterly J. Math., 13, 1942,
    30-39.
[3] A.E. Ingham, The distribution of prime numbers. Cambridge Tracts, 30, Cambridge 1932.
[4] X.-J. Li, The positivity of a sequence of numbers and the Riemann Hypothesis. J. Number Theory, 65,
    1997, 325-333.
remarks on weil’s quadratic functional in the theory of prime numbers, I                                233


[5] H. Yoshida, On Hermitian Forms attached to Zeta Functions. In: N. Kurokawa - T. Sunada (eds.),
    Zeta Fuctions in Geometry. Advanced Studies in Pure Mathematics, 21, Mathematical Society of Japan,
    Kinokuniya, Tokyo 1992, 281-325.
[6] A. Weil, Sur les “formules explicites’’ de la théorie des nombres premiers. Meddelanden Från Lunds Univ.
    Mat. Sem. (dedié a M. Riesz), 1952, 252-265.
[7] E.T. Whittaker - G.N. Watson, A Course of Modern Analysis. Fourth edition, Cambridge University
    Press, Cambridge 1952.


Pervenuta il 15 giugno 2000,
in forma definitiva il 7 settembre 2000.

                                                                              Institute for Advanced Study
                                                                            Princeton, NJ 08540 (U.S.A.)
                                                                                                eb@ias.edu
