                                                  A STRENGTHENING OF THE NYMAN-BEURLING
                                                   CRITERION FOR THE RIEMANN HYPOTHESIS




arXiv:math/0202141v2 [math.NT] 18 Feb 2002
                                                                            LUIS BÁEZ-DUARTE


                                                    Abstract. Let ρ(x) = x − [x], χ = χ(0,1) . In L2 (0, ∞) consider     the
                                                                                                                1
                                                                                                                  
                                                    subspace B generated by {ρa |a ≥ 1} where ρa (x) := ρ ax        . By the
                                                    Nyman-Beurling criterion the Riemann hypothesis is equivalent to the
                                                    statement χ ∈ B. For some time it has been conjectured, and proved
                                                    in this paper, that the Riemann hypothesis is equivalent to the stronger
                                                    statement that χ ∈ Bnat where Bnat is the much smaller subspace gen-
                                                    erated by {ρa |a ∈ N}.




                                                                            1. Introduction
                                                We denote the fractional part of x by ρ(x) = x − [x], and let χ stand
                                             for the characteristic function of the interval (0, 1]. µ denotes the Möbius
                                             function. We shall be working in the Hilbert space

                                                                             H := L2 (0, ∞),
                                             where the main object of interest is the subspace of Beurling functions,
                                             defined as the linear hull of the family {ρa |1 ≤ a ∈ R} with
                                                                                         
                                                                                          1
                                                                            ρa (x) := ρ       .
                                                                                         ax
                                             The much smaller subspace B nat of natural Beurling functions is generated
                                             by {ρa |a ∈ N}. The Nyman-Beurling criterion ([13], [6]) states, in a slightly
                                             modified form [4] (the original formulation is related to L2 (0, 1)), that the
                                             Riemann hypothesis is equivalent to the statement that

                                                                                    χ ∈ B,

                                             but it has recently been conjectured by several authors1 that this condition
                                             could be substituted by χ ∈ B nat . We state this as a theorem to be proved
                                             below.
                                             Theorem 1.1. The Riemann hypothesis is equivalent to the statement that

                                                                                  χ ∈ B nat .
                                               Date: 18 February 2002.
                                               Key words and phrases. Riemann hypothesis, Nyman-Beurling theorem.
                                               1
                                                 see [1], [2], [3], [4], [5], [8], [9], [10], [11], [12], [16], [17]
                                                                                       1
2                              LUIS BÁEZ-DUARTE



To properly gauge the strength of this theorem note this: not only is B nat
a rather thin subspace of B but, as is easily seen, it is also true that B is
much larger than B nat .
  By necessity all authors have been led in one way or another to the natural
approximation
                                       n
                                       X
(1.1)                          Fn :=          µ(a)ρa ,
                                       a=1

which tends to −χ both a.e. and in L1 norm when restricted to (0, 1) (see
[1]), but which has been shown ([2], [3]) to diverge in H. In unpublished
work one has tried to prove convergence under the Riemann hypothesis of
subsequences
   Pn         of {Fn } such as when n is restricted to run along the solutions
of a=1 µ(n) = 0. Another attempt by J. B. Conrey and G. Myerson [8]
relates to a mollification of Fn , the Selberg approximation, defined in [4] by
                               n                        
                               X              log a
                       Sn :=         µ(a) 1 −                 ρa .
                                              log n
                               a=1

A common problem to these sequences is that if they converge atP
                                                               all to −χ in
H they must do so very slowly: it is known [4] that for any F = nk=1 ck ρak ,
ak ≥ 1, if N = max ak , then

                                                  C
(1.2)                       kF − χkH ≥ √               ,
                                                 log N

for an absolute constant C that has recently been sharpened by J. F. Burnol
[7]. This, as well as considerations of summability of series, led the author
in [3] as well as here to try to employ symultaneously, as it were, the whole
range of a ∈ [1, ∞). Thus we define for complex s and x > 0 the functions

                                       ∞
                                       X µ(a)
(1.3)                      fs (x) :=                ρa (x).
                                               as
                                       a=1
For fixed x > 0 this is a meromorphic functions of s in the complex plane
since

                               1       X µ(a)  1 
                  fs (x) =           −                 ,
                           xζ(s + 1)         as ax
                                              a≤1/x
where the finite sum on the right is an entire function; thus fs is seen to be
a sort of correction of 1/ζ(s). Assuming the Riemann hypothesis we shall
prove for small positive ǫ that

                                     fǫ ∈ B nat ,
                         ON NYMAN-BEURLING CRITERION                               3

and then, unconditionally, that
                                       H
                                    fǫ → −χ,          (ǫ ↓ 0),
so that χ ∈ B nat .

                                     2. The Proof
2.1. Two technical lemmae. Here s = σ + iτ with σ and τ real. The
well-known theorem of Littlewood (seeP[15] Theorem   14.25 (A)) to the effect
that under the Riemann hypothesis ∞    a=1 µ(a)a−s converges to 1/ζ(s) for

ℜ(s) > 1/2 has been provided in the more general setting of ℜ(s) > α with
a precise error term by M. Balazard and E. Saias ([5], Lemme 2). We quote
their lemma here for the sake of convenience.
Lemma 2.1. Let 1/2 ≤ α < 1, δ > 0, and ǫ > 0. If ζ(s) does not vanish in
the half-plane ℜ(s) > α, then for n ≥ 2 and α + δ ≤ ℜ(s) ≤ 1 we have

                  n
                  X µ(a)             1                             
(2.1)                           =        + Oα,δ,ǫ n−δ/3 (1 + |τ |)ǫ
                  a=1
                         as         ζ(s)

It is important to note that the next lemma is independent of the Riemann
or even the Lindelöf hypothesis.
Lemma 2.2. For 0 ≤ ǫ ≤ ǫ0 < 1/4 there is a positive constant C = C(ǫ0 )
such that for all τ
                              ζ( 12 − ǫ + iτ )
(2.2)                            1             ≤ C (1 + |τ |)ǫ .
                              ζ( 2 + ǫ + iτ )
Proof. We bring in the functional equation of ζ(s) to bear as follows
                  ζ( 12 − ǫ + iτ )                  ζ( 12 − ǫ − iτ )
                                           =
                  ζ( 21 + ǫ + iτ )                  ζ( 21 + ǫ + iτ )
                                                        Γ( 14 + 21 ǫ + 12 iτ )
                                           = π −ǫ                              ,
                                                        Γ( 14 + 21 ǫ + 12 iτ )
then the conclusion easily follows from well-known asymptotic formulae for
the gamma function in a vertical strip ([14] (21.51), (21.52)).
2.2. The proof proper of Theorem 1.1. It is clear that we need not
prove the if part of Theorem 1.1. So let us assume that the Riemann
hypothesis is true. We define
                                      n
                                      X µ(a)
                          fǫ,n :=                    ρa ,   (ǫ > 0).
                                      a=1
                                               aǫ
It is easy to see that
4                              LUIS BÁEZ-DUARTE



                                 n        n     
                               1 X µ(a) X µ(a) 1
(2.3)               fǫ,n (x) =          −          ,
                               x   a1+ǫ     aǫ ax
                                  a=1            a=1

then, noting that the terms of the right-hand sum drop out when a > 1/x,
we obtain the pointwise limit

                                       1       X µ(a)  1 
(2.4)      fǫ (x) = lim fǫ,n (x) =           −              .
                   n→∞             xζ(1 + ǫ)      aǫ ax
                                                         a≤1/x

Then again for fixed x > 0 we have
                                         
                                 X        1
(2.5)             lim fǫ (x) = −   µ(a)     = −χ(x),
                   ǫ↓0                   ax
                                   a≤1/x

by the fundamental property on Möbius numbers. The task at hand now is
to prove these pointwise limits are also valid in the H-norm. To this effect
we introduce a new Hilbert space

                         K := L2 ((∞, ∞), (2π)−1/2 dt),

and note that by virtue of Plancherel’s theorem the Fourier-Mellin map M
defined by
                                        Z ∞
                                                 1
(2.6)                  M(f )(τ ) :=           x− 2 +iτ f (x)dx,
                                        0

is an invertible isometry from H to K. A well-known identity, which is at
the root of the Nyman-Beurling formulation, probably due to Titchmarsh
([15], (2.1.5)), namely
                        Z ∞
                 ζ(s)
              −       =      xs−1 ρ1 (x)dx, (0 < ℜ(s) < 1),
                  s      0
immediately yields, denoting Xǫ (x) = x−ǫ ,

                                                     n
                               ζ( 12 − ǫ + iτ ) X µ(a)
(2.7)   M(Xǫ f2ǫ,n )(τ ) = −     1                1       ,       (0 < ǫ < 1/2).
                                 2 − ǫ + iτ a=1 a 2
                                                    +ǫ+iτ


By a theorem of Littlewood ([15], Theorem 14.25 (A)) if we let n → ∞ in
the right-hand side of (2.7) we get the pointwise limit

                              n
               ζ( 12 − ǫ + iτ ) X µ(a)        ζ( 12 − ǫ + iτ )    1
(2.8)      −     1                1       → −     1            1         .
                 2 − ǫ + iτ a=1 a 2
                                    +ǫ+iτ     ζ( 2 + ǫ + iτ ) 2 − ǫ + iτ
                        ON NYMAN-BEURLING CRITERION                             5



To see that this limit also takes place in H we choose the parameters in
Lemma 2.1 as α = 1/2, δ = ǫ > 0, ǫ ≤ 1/2, and n ≥ 2 to obtain
               n
               X µ(a)                      1
                       1         =      1            + Oǫ ((1 + |τ |)ǫ ) .
               a=1 a   2
                         +ǫ+iτ       ζ( 2 + ǫ + iτ )
If we now use Lemma 2.2 and the Lindelöf hypothesis applied to the abcissa
1/2 − ǫ, which follows from the Riemann hypothesis, we obtain a positive
constant Kǫ such that for all real τ
                                      n
                  ζ( 21 − ǫ + iτ ) X µ(a)
              −     1                    1       ≤ Kǫ (1 + |τ |)−1+2ǫ .
                    2  − ǫ + iτ    a=1 a 2
                                           +ǫ+iτ


It is then clear that for 0 < ǫ < 1/4 the left-hand side of (2.8) is uniformly
majorized by a function in K. Thus the convergence does take place in K
which implies that

                                                  H
                                     Xǫ f2ǫ,n → Xǫ f2ǫ .

But x−ǫ > 1 for 0 < x < 1, and for x > 1

                                          n
                                     1 X µ(a)    1
(2.9)               f2ǫ,n (x) =             1+ǫ
                                                ≪ ,           (x > 1),
                                     x a=1 a     x
which easily implies that one also has H-convergence for f2ǫ,n as n → ∞. The
factor 2 in the subindex is unessential, so that we now have for sufficiently
small ǫ > 0 that

                                              H
                                     fǫ,n → fǫ ∈ B nat ,

as was announced above. Moreover, since we have identified the pointwise
limit in (2.8) we now have

                                              ζ( 21 − ǫ + iτ )    1
                  M(Xǫ f2ǫ )(t) = −              1             1         .
                                              ζ( 2 + ǫ + iτ ) 2 − ǫ + iτ

Now we apply Lemma 2.2 and obtain, without the assumption of the Rie-
mann hypothesis, that M(Xǫ f2ǫ ) converges in K, thus Xǫ f2ǫ converges in H,
and this means that fǫ also converges in H as ǫ ↓ 0 by an argument entirely
similar to that used for fǫ,n. The identification of the pointwise limit in (2.5)
finally gives

                                                  H
                                          fǫ → −χ,
6                                LUIS BÁEZ-DUARTE



which concludes the proof.

                    3. Some comments and a corollary
   The proof of Theorem 1.1 provides in turn a new proof, albeit of a stronger
theorem, of the Nyman-Beurling criterion which bypasses the deep and com-
plicated Hardy space techniques. One should extend it to the Lp case, that
is, to the condition that ζ(s) does not vanish in a half-plane ℜ(s) > 1/p. It
should be clear also that we have shown this special equivalence criterion to
be true:
Corollary 3.1. The Riemann hypothesis is equivalent to the H-convergence
of fǫ,n as n → ∞ for all sufficiently small ǫ > 0.
   In essence what has been done is to apply a summability method to the
old natural approximation. The convergence on special subsequences both
of n and of ǫ is also necessary and sufficient, and it is proposed here to study
this alongside with other summability methods for the natural approxima-
tion.
A final remark is in order. Note that we did not employ the dependence
on n in the Balazard-Saias Lemma 2.1. This dependence would seem to be
closely connected to the slowness of approximation to −χ indicated in (1.2).

Ackowledgements. The author wants to thank M. Balazard and E. Saias
for pointing out their important, delicate lemmae 2.1.

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Luis Báez-Duarte
Departamento de Matemáticas
Instituto Venezolano de Investigaciones Cientı́ficas
Apartado 21827, Caracas 1020-A
Venezuela

   E-mail address: lbaez@ccs.internet.ve
