                                               A NOTE ON SOME POSITIVITY CONDITIONS RELATED TO
                                                            ZETA AND L-FUNCTIONS


                                                                       J. B. Conrey and Xian-Jin Li




arXiv:math/9812166v1 [math.NT] 3 Dec 1998
                                                                                1. Introduction
                                               The theory of Hilbert spaces of entire functions [1] was developed by Louis de
                                            Branges in the late 1950s and early 1960s. It is a generalization of the part of
                                            Fourier analysis involving Fourier transforms and the Plancherel formula. In [2] de
                                            Branges proposed an approach to the generalized Riemann hypothesis, that is, the
                                            hypothesis that not only the Riemann zeta function ζ(s) but also all the Dirichlet L-
                                            functions L(s, χ) with χ primitive have their nontrivial zeros lying on the critical line
                                            ℜs = 1/2 (See Davenport [5]). In [2] de Branges mentioned that his approach to the
                                            generalized Riemann hypothesis using Hilbert spaces of entire functions is related
                                            to the Lax-Phillips theory of scattering [6]. In Appendix 2 to Section 7, [6] Lax and
                                            Phillips explained the difficulty of approaching the Riemann hypothesis by using
                                            the scattering theory. In this note, we shall indicate the difficulty of approaching
                                            the Riemann hypothesis by using de Branges’ positivity conditions [2] [3] [4]. In
                                            fact, we shall give examples showing that de Branges’ positivity conditions, which
                                            imply the generalized Riemann hypothesis, are not satisfied by defining functions of
                                            reproducing kernel Hilbert spaces associated with the Riemann zeta function ζ(s)
                                            and the Dirichlet L-function L(s, χ4 ).

                                                                2. Reproducing kernel Hilbert spaces
                                               We first outline an important part of de Branges’ approach to the Riemann
                                            hypothesis.
                                               Let E(z) be an entire function satisfying |E(z̄)| < |E(z)| for z in the upper half-
                                            plane. A Hilbert space of entire functions H(E) is the set of all entire functions
                                            F (z) such that F (z)/E(z) is square integrable on the real axis and such that

                                            (2.1)                          |F (z)|2 ⩽ kF k2H(E) K(z, z)

                                            for all complex z, where the inner product of the space is given by
                                                                                              Z ∞
                                                                                                F (x)Ḡ(x)
                                                                     hF (z), G(z)iH(E) =                2
                                                                                                           dx
                                                                                              −∞ |E(x)|

                                              1991 Mathematics Subject Classification. 11M26.
                                              Key words and phrases. Zeros of zeta and L-functions, Hilbert spaces of entire functions.
                                              Research of both authors supported by the American Institute of Mathematics.

                                                                                                                     Typeset by AMS-TEX
                                                                                          1
2                          J. B. CONREY AND XIAN-JIN LI

for all elements F, G ∈ H(E) and where

                                      E(z)Ē(w) − Ē(z̄)E(w̄)
                          K(w, z) =
                                           2πi(w̄ − z)
is the reproducing kernel function of the space H(E), that is, the identity

(2.2)                        F (w) = hF (z), K(w, z)iH(E)

holds for every complex w and for every element F ∈ H(E). The identity (2.2) is
obtained by using Cauchy’s integration formula in the upper half-plane (cf. [1]),
and the condition (2.1) is made so that Cauchy’s formula applies to all functions in
the space H(E).
   The following two theorems are essentially due to de Branges (cf. [2] [3]).
Theorem 1. Let E(z) be an entire function having no real zeros such that |E(z̄)| <
|E(z)| for ℑz > 0, such that Ē(z̄) = ǫE(z −i) for a constant ǫ of absolute value one,
and such that |E(x + iy)| is a strictly increasing function of y > 0 for each fixed real
x. If ℜhF (z), F (z + i)iH(E) ⩾ 0 for every element F (z) ∈ H(E) with F (z + i) ∈
H(E), then the zeros of E(z) lie on the line ℑz = −1/2, and ℜ{Ē ′ (w)E(w +
i)/2πi} ⩾ 0 when w is a zero of E(z).
Proof. Let w be a zero of E(z). Since Ē(z̄) = ǫE(z − i) with |ǫ| = 1, we have

(2.3)              Ē(w + i)K(w, z + i) = −Ē(w − i)K(w + i, z)

for all complex z. Since E(z) has no real zeros and since |E(z̄)| < |E(z)| for z in
the upper half-plane, E(w + i) and E(w − i) are nonzero. It follows that K(w, z)
is a nonzero element of H(E) such that K(w, z + i) belongs to the space. Assume
that F (z) is an element in H(E) such that F (z + i) belongs to the space. Then, by
(2.2) and (2.3), we have
                 hF (z + i), K(w, z)iH(E) + hF (z), K(w, z + i)iH(E)
(2.4)                 E(w + i) − E(w − i)
                  =                       F (w + i).
                           E(w + i)
Define a new scalar product h·, ·i by

            hF (z), G(z)i = hF (z + i), G(z)iH(E) + hF (z), G(z + i)iH(E)

for all F, G ∈ H(E) such that F (z + i), G(z + i) ∈ H(E). Since, by assumption,
ℜhF (z+i), F (z)iH(E) ⩾ 0 for every element F (z) ∈ H(E) such that F (z+i) ∈ H(E),
we have hF (z), F (z)i ⩾ 0. Then, by (2.4) and the Schwarz inequality, we have
                                                 2
                E(w + i) − E(w − i)
                                      F (w + i) = |hF (z), K(w, z)i|2
                       E(w + i)
(2.5)
                ⩽ hF (z), F (z)ihK(w, z), K(w, z)i
                = 4Re K(w, w + i)Re hF (z + i), F (z)iH(E) .
                              ZETA AND L-FUNCTIONS                                    3

   If K(w, w + i) 6= 0, then we must have ℑw = −1/2 because, otherwise, we have
K(w, w + i) = 0 by the functional identity Ē(z̄) = ǫE(z − i).
   Next, we assume that K(w, w + i) = 0. Then F (z) = K(w, z) is an element of
H(E) such that F (w + i) = 0 and F (z + i) ∈ H(E). If F (z) is a nonzero element
of H(E) having zero at a point z0 , it is easy to see by definition that F (z)/(z − z0 )
belongs to H(E). Since F (z) is an entire function, by using Taylor’s expansion of
F (z) at the point z0 , we see that F (z)/(z − z0 )n does not vanish at z0 for some
positive integer n. By the repeated process of dividing out the factor z − z0 from
F (z), we see that F (z)/(z − z0 )n belongs to H(E). If F (z + i) ∈ H(E), we also
see that F (z + i)/(z + i − z0 )n ∈ H(E). Therefore, there exists a nonzero element
F (z) ∈ H(E) such that F (w+i) 6= 0 and F (z+i) ∈ H(E). Hence, if K(w, w+i) = 0,
then, by (2.5), we have
                              E(w + i) − E(w − i) = 0.
Since Ē(w̄) = ǫE(w − i), we have |E(w + i)| = |E(w̄)|. Note that ℜ(w + i) = ℜ(w̄).
Since |E(x + iy) is a strictly increasing function of y on (0, ∞), we must have
ℑ(w + i) = ℑ(w̄), and hence w + i = w̄. Therefore, we have ℑw = − 12 .
   We have K(w, w + i) = Ē ′ (w)E(w + i)/2πi. Since F (z) = K(w, z) is an element
of H(E) such that F (z + i) ∈ H(E), we have
                    ℜK(w, w + i) = ℜhF (z + i), F (z)iH(E) ⩾ 0,
that is, ℜ{Ē ′ (w)E(w + i)/2πi} ⩾ 0 when w is a zero of E(z).
   This completes the proof of the theorem.
   Let W (z) be a function analytic and having no zeros in the upper half-plane.
Then a Hilbert space of analytic functions F (W ) is the set of all analytic functions
F (z) in the upper half-plane, such that F (z)/W (z) can be written as a quotient of
bounded analytic functions in the upper half-plane, has square integrable boundary
values on the real axis, and satisfies the inequality
                                              y +∞ log |F (t)/W (t)|dt
                                               Z
              log |F (x + iy)/W (x + iy)| ⩽
                                              π −∞     (t − x)2 + y 2
for y > 0. The inner product of F (W ) is given by
                                           Z ∞
                                                F (x)Ḡ(x)
                      hF (z), G(z)iF(W ) =              2
                                                           dx
                                            −∞ |W (x)|

for all F, G ∈ F (W ). The reproducing kernel function of F (W ) is given by the
expression
                                      W (z)W̄ (w)
                            K(w, z) =              ,
                                       2πi(w̄ − z)
that is, for every complex w in the upper half-plane, we have
(2.6)                       F (w) = hF (z), K(w, z)iF(W )
for every element F ∈ F (W ). The identity (2.6) is obtained by using Cauchy’s
integration formula in the upper half-plane (cf. [1]).
4                           J. B. CONREY AND XIAN-JIN LI

Theorem 2. Let W (z) be a function analytic and having no zeros in the upper half-
plane. Let T be a linear transformation of F (W ) into itself which takes K(w, z)
into K(w + i, z) for all complex w with ℑw > 0. Assume that

                                ℜhF (z), T F (z)iF(W ) ⩾ 0

for all F ∈ F (W ). Then W (z) has an analytic extension to the half-plane ℑz >
−1/2, and W (z)/W (z + i) has a nonnegative real part in this half-plane.
Proof. Let w1 , . . . , wr be points
                              Pr in the upper half-plane, andP
                                                             let c1 , . . . , cr be com-
                                                               r
plex numbers. If F (z) = α=1 cα K(wα , z), then T F (z) = β=1 cβ K(wβ + i, z).
By assumption, we have
      r
      X
             cα c̄β [K(wα , wβ + i) + K(wα + i, wβ )] = 2ℜhF (z), T F (z)iF(W ) ⩾ 0,
     α,β=1


that is, the expression K(w + i, z) + K(w, z + i) is positive-definite for w, z in the
upper half-plane. This implies that ℜ{W (z)/W (z + i)} ⩾ 0 for z in the upper
half-plane. Let
                                     W (z) − W (z + i)
                            B(z) =                     .
                                     W (z) + W (z + i)
Then B(z) is analytic and bounded by one in the upper half-plane. The positive-
definiteness of K(w + i, z) + K(w, z + i) implies the positive-definiteness of the
expression
                                 1 − B(z)B̄(w)
                                2πi(w̄ − z − i)
for w, z in the upper half-plane.
   Let H be the Hilbert space of analytic functions in the half-plane ℑz > −1/2,
which has the expression L(w, z) = 1/2πi(w̄ − z − i) as its reproducing kernel
function. The norm of an element F in the space H is given by
                                    Z ∞
                   hF (z), G(z)iH =     F (x − i/2)Ḡ(x − i/2)dx.
                                        −∞

  Let P be a transformation of H into itself, which takes L(w, z) into B̄(w)L(w, z).
The positive-definiteness of the expression

                                     1 − B(z)B̄(w)
                                     2πi(w̄ − z − i)

implies hP F (z), P F (z)iH ⩽ hF (z), F (z)iH for all elements F ∈ H which are linear
combination of functions L(w, z) with ℑw > 0. Since L(w, z) is the reproducing
kernel function of H, if F ∈ H is orthogonal to all elements L(w, z) with ℑw > 0,
then
                             F (w) = hF (z), L(w, z)iH = 0
                             ZETA AND L-FUNCTIONS                                        5

for ℑw > 0. Since F (z) is analytic for ℑz > −1/2, we must have F ≡ 0. There-
fore, the set of elements L(w, z) with ℑw > 0 is dense in H. It follows that
hP F (z), P F (z)iH ⩽ hF (z), F (z)iH for all elements F ∈ H. Thus, P is a bounded
linear transformation of the Hilbert space H into itself, and therefore, the adjoint
P ∗ of P exists.
   Let α be a complex number with ℑα > −1/2, and let F (z) = L(α, z). Then
F ∈ H, and hence P ∗ F (z) ∈ H. It follows that
                                               Z ∞
                         ∗                                               ᾱ − w − i
        2πi(ᾱ − w − i)hP F (z), L(w, z)iH =           P ∗ F (x − i/2)              dx
                                                  −∞                     x − 2i − w

is an analytic function of w for ℑw > −1/2. Since

  2πi(ᾱ − w − i)hP ∗ F (z), L(w, z)iH = 2πi(ᾱ − w − i)hF (z), P L(w, z)iH = B(w)

for w in the upper half-plane, B(z) has an analytic extension to the half-plane
ℑz > −1/2.
   If F ∈ H and ℑw > 0, we have

(2.7)                    B(w)F (w) = hP ∗ F (z), L(w, z)iH .

Since both sides of (2.7) are analytic functions of w for ℑw > −1/2, the identity
(2.7) remains true for all complex w with ℑw > −1/2 by analytic continuation.
Since hP F (z), P F (z)iH ⩽ hF (z), F (z)iH for all F ∈ H, we have

                |B(w)F (w)|2 = |hF (z), P L(w, z)iH|2
                             ⩽ hF (z), F (z)iH hP L(w, z), P L(w, z)iH
(2.8)
                             ⩽ hF (z), F (z)iH hL(w, z), L(w, z)iH
                             = hF (z), F (z)iH L(w, w).

for ℑw > −1/2 and for all F ∈ H. In particular, if F (z) = L(w, z), then
hF (z), F (z)iH = F (w), and hence (2.8) becomes |B(w)F (w)|2 ⩽ |F (w)|2 for ℑw >
−1/2, that is, |B(w)| ⩽ 1 for ℑw > −1/2.
   Therefore, we have proved that B(z) is analytic and bounded by one for ℑz >
−1/2. It follows that W (z)/W (z + i) is analytic and has nonnegative real part in
the half-plane ℑz > −1/2.
   This completes the proof of the theorem.

             3. Hilbert spaces associated with ζ(s) and L(s, χ4 )
3.1. The Riemann zeta function. The Riemann zeta function ζ(s) is given by
                                            ∞
                                            X 1
                                   ζ(s) =
                                            n=1
                                                  ns
6                          J. B. CONREY AND XIAN-JIN LI

for ℜs > 1. Let ξ(s) = s(s − 1)π −s/2 Γ( 2s )ζ(s). Then ξ(s) is an entire function, and
satisfies the functional identity ξ(s) = ξ(1 − s). It is well-known (See Davenport
[5]) that we have the infinite product formula
                                        Y          s
                                                      
                                 ξ(s) =        1−
                                                    ρ

where the product is taken over all nontrivial zeros ρ of ζ(s) with ρ and 1 − ρ being
paired together for the convergence of the product.
   Let E(z) = ξ(1 − iz). Then the Riemann hypothesis is that the zeros of E(z) lie
on the line ℑz = −1/2, and the functional identity ξ(s) = ξ(1 − s) can be written as
Ē(z̄) = E(z − i). If ρ is a nontrivial zero of ζ(s), then 0 < ℜρ < 1, a result proved
by Hadamard and de la Vallée Poussin independently in 1896. Since
                                       2       Y (ℜρ + y)2 + (ℑρ − x)2
                      2
                           Y      iz
                |E(z)| =       1−          =
                                  ρ                         |ρ|2

for z = x + iy, we see that |E(x − iy)| < |E(x + iy)| for y > 0, and that |E(x + iy)|
is a strictly increasing function of y on (0, ∞) for each fixed real x.
   In view of Theorem 1, it is natural to ask whether the Hilbert space of entire
functions H(E) satisfies the condition that

(3.1)                        ℜhF (z), F (z + i)iH(E) ⩾ 0

for every element F (z) of H(E) such that F (z + i) ∈ H(E), because the nontrivial
zeros of the Riemann zeta function ζ(s) would then lie on the critical line ℜs = 1/2
under this condition. In the following, we give an example showing that condition
(3.1) is unfortunately not true.
   Let ρ = 1/2 + i111.0295355431696745 · · · be the 34th zero of the Riemann zeta
function in the upper half-plane. By using MATHEMATICA, we compute that

(3.2)       −ℜ{ξ ′ (ρ)ξ(1 + ρ)} = −5.389100507182945 · · · × 10−69 < 0.

Write ρ = 1 − iw. Then E(w) = 0, and Ē ′ (w)E(w + i)/i = −ξ ′ (ρ)ξ(1 + ρ). Thus,
(3.2) becomes
                        ℜ{Ē ′ (w)E(w + i)/2πi} < 0.
Therefore, by Theorem 1, we see that the Hilbert space of entire functions H(E)
with E(z) = ξ(1 − iz) does not satisfy the condition (3.1).
   Next, let W (z) = 1/ξ(1 − iz). Then W (z) is analytic in the upper half-plane,
and is continuous and having no zeros in the closed upper half-plane. If the Hilbert
space of analytic functions F (W ) satisfy the condition that

(3.3)                          ℜhF (z), T F (z)iF(W ) ⩾ 0

for all F ∈ F (W ), where T is the linear transformation of F (W ) into itself which
takes K(w, z) = W (z)W̄ (w)/2πi(w̄ − z) into K(w + i, z) for all complex w with
                             ZETA AND L-FUNCTIONS                                     7

ℑw > 0, then the function W (z) would have analytic extension to the half-plane
ℑz > −1/2 by Theorem 2, that is, the Riemann zeta function ζ(s) has no zeros for
ℜs > 1/2. In the following, we give an example showing that the space F (W ) does
not satisfy the condition (3.3).
  By using MATHEMATICA, we compute that

(3.4)            ℜ{ξ(1 + i282)/ξ(2 + i282)} = −0.000131957 < 0.

Let w = −282. Then ℑw = 0 > −1/2. Since W (w) = 1/ξ(1 +i282) and W (w +i) =
1/ξ(2 + i282), by (3.4) we have

                              ℜ{W (w)/W (w + i)} < 0.

Therefore, by Theorem 2, we see that the space F (W ) with W (z) = 1/ξ(1 − iz)
does not satisfy the condition (3.3).
3.2. The Dirichlet L-function L(s, χ4 ). Note that χ4 is the real primitive Dirich-
let character (mod 4), which is given by
                                           n−1
                                        (−1) 2 ,    if n is odd;
                         χ4 (n) =
                                        0,          if n is even.

The Dirichlet L-function L(s, χ4 ) is given by
                                              ∞
                                              X χ4 (n)
                                L(s, χ4 ) =
                                              n=1
                                                     ns

for ℜs > 0. Let ξ(s, χ4 ) = (4/π) 2 Γ( 1+s
                                    s
                                        2 )L(s, χ4 ). Then ξ(s, χ4 ) is an entire func-
tion, and satisfies the functional identity

(3.5)                        ξ(1 − s, χ4 ) = ǫ(χ4 )ξ(s, χ4 )

where ǫ(χ4 ) is a constant of absolute value one (See Davenport [5]). Since χ4 is a
real character, by the argument of section 12 of Davenport [5], we have the infinite
product formula
                                         √ Y         
                                           π        s
(3.6)                        ξ(s, χ4 ) =        1−
                                          2         ρ

where the product is taken over all nontrivial zeros of L(s, χ4 ) with ρ and 1 − ρ
being put together.
   Let Eχ4 (z) = ξ(1 − iz, χ4 ). Then the functional identity (3.5) can be written as

                             Ēχ4 (z̄) = ǭ(χ4 )Eχ4 (z − i).

By the product formula (3.6), we find that |Eχ4 (x + iy)| is a strictly increasing
function of y > 0 for each fixed real x. Since the nontrivial zeros of L(s, χ4 ) lie
8                          J. B. CONREY AND XIAN-JIN LI

in the strip 0 < ℜs < 1 (See §14, Davenport [5]), we have |Eχ4 (z̄)| < |Eχ4 (z)| for
ℑz > 0.
   In view of Theorem 1, it is natural to ask whether the Hilbert space of entire
functions H(Eχ4 ) satisfies the condition that

(3.7)                        ℜhF (z), F (z + i)iH(Eχ4 ) ⩾ 0

for every element F (z) of H(Eχ4 ) such that F (z + i) ∈ H(Eχ4 ), because the non-
trivial zeros of the Dirichlet L-function L(s, χ4 ) would then lie on the critical line
ℜs = 1/2 under this condition. In the following, we give an example showing that
condition (3.7) is unfortunately not true.
   Let ρ = 1/2 + i67.6369208635460683980549 · · · be a zero of L(s, χ4 ). By using
MATHEMATICA, we compute that

        −ℜ{ξ ′ (ρ, χ4 )ξ(1 + ρ, χ4 )} = −2.310349004993483456 · · · × 10−45 < 0.

Write ρ = 1 − iw. Then Eχ4 (w) = 0, and E¯χ′ 4 (w)Eχ4 (w + i)/i = −ξ ′ (ρ, χ4 )ξ(1 +
ρ, χ4 ). Thus, the above inequality becomes

                           ℜ{E¯χ′ 4 (w)Eχ4 (w + i)/2πi} < 0.

Therefore, by Theorem 1, we see that the Hilbert space of entire functions H(Eχ4 )
with Eχ4 (z) = ξ(1 − iz, χ4 ) does not satisfy the condition (3.7).
   Next, let Wχ4 (z) = 1/ξ(1 − iz, χ4 ). Then Wχ4 (z) is analytic in the upper half-
plane, and is continuous and having no zeros in the closed upper half-plane. If the
space F (Wχ4 ) satisfy the condition that

(3.8)                         ℜhF (z), T F (z)iF(Wχ4 ) ⩾ 0

for all F ∈ F (Wχ4 ), where T is the linear transformation of F (Wχ4 ) into itself
which takes K(w, z) = Wχ4 (z)W̄χ4 (w)/2πi(w̄ − z) into K(w + i, z) for all complex
w with ℑw > 0, then the function Wχ4 (z) would have analytic extension to the
half-plane ℑz > −1/2 by Theorem 2, that is, the Dirichlet L-function L(s, χ4 ) has
no zeros for ℜs > 1/2. In the following, we give an example showing that the space
F (Wχ4 ) does not satisfy the condition (3.8).
   By using MATHEMATICA, we compute that

         ℜ{ξ(1 + i8714.2, χ4 )/ξ(2 + i8714.2, χ4 )} = −0.000422340607 < 0.

Let w = −8714.2. Then ℑw = 0 > −1/2. Since Wχ4 (w) = 1/ξ(1 + i8714.2, χ4 ) and
Wχ4 (w + i) = 1/ξ(2 + i8714.2, χ4 ), we have

                            ℜ{Wχ4 (w)/Wχ4 (w + i)} < 0.

Therefore, by Theorem 2, we see that the space F (Wχ4 ) with Wχ4 (z) = 1/ξ(1 −
iz, χ4 ) does not satisfy the condition (3.8).
                                ZETA AND L-FUNCTIONS                                          9

                                     4. Conclusion
   We have seen in section 3 the difficulty of approaching the generalized Riemann
hypothesis by using de Branges type positivity conditions for reproducing kernel
Hilbert spaces. It is possible that these positivity conditions are too strong for
Hilbert spaces of entire functions associated with the Riemann zeta function and
the Dirichlet L-functions.

Remark. After he looked at the manuscript of this paper, Peter Sarnak gave a
proof for the statement that the space F (W ) does not satisfy the condition (3.3)
where W (z) = 1/ξ(1 − iz), and his argument involves no numberical calculations.
For the convenience of readers, we sketch his proof here. Let F (s) = ξ(s)/ξ(s + 1).
Assume that all logarithmic functions are defined by continuation from the point
s = 2 at which their arguments are set to be zero. Then we have

                           ℑ{log F (s)} = ℑ{log ζ(s)} + O(1)

for ℜs > 1/2. Since the set of values of log ζ(s), 1/2 < ℜs < 2, is dense in the
complex plane (See Chapter XI of Titchmarsh [7]), a complex number s0 exists
with ℜs0 > 1/2 such that π/2 < ℑ{log F (s0 )} < π. Let z0 = i(s0 − 1). Then
ℑz0 > −1/2 and ℜ{W (z0 )/W (z0 + i)} < 0, and hence by Theorem 2 the space
F (W ) does not satisfy the condition (3.3). Let r be any positive integer. For any
Dirichlet character χ modulo r, let
                                              s+a   s+a
                          ξ(s, χ) = (π/r)− 2 Γ(         )L(s, χ)
                                                     2
where L(s, χ) is the Dirichlet L-function and where a = 0 if χ(−1) = 1 and a = 1 if
χ(−1) = −1. By using a similar argument, Peter Sarnak also proved that the space
F (Wχ ) does not satisfy the condition (3.8) where Wχ (z) = 1/ξ(1 − iz, χ).

                                       References
1. L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, 1968.
2. L. de Branges, The Riemann hypothesis for Hilbert spaces of entire functions, Bull. Amer.
   Math. Soc. 15 (1986), 1–17.
3. L. de Branges, The convergence of Euler products, J. Functional Analysis 107 (1992), 122–210.
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   121 (1994), 117–184.
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                                         Appendix
   Here is a list of MATHEMATICA instructions, which can be used to verify our
calculations.
10                        J. B. CONREY AND XIAN-JIN LI

     xi[s ]:=s*(s-1)*Pi^ (-s/2)*Gamma[s/2]*Zeta[s]
     f[ro ]:=Re[-xi’[ro]*xi[ro+1]]
     g[t ]:=Re[xi[1+I*t]/xi[2+I*t]]
     ro1=1/2+I*111.029535543169674524656
     Zeta[ro1]
     f[ro1]
     g[282.]
     Plot[g[t],{t,281.95,282.15}]
     xi4[s ]:=(4*Pi)^ (-s/2)*Gamma[(s+1)/2]*(Zeta[s,1/4]-Zeta[s,3/4])
     f4[ro ]:=Re[-xi4’[ro]*xi4[ro+1]]
     g4[t ]:=Re[xi4[1+I*t]/xi4[2+I*t]]
     ro2=1/2+I*67.6369208635460683980549
     Zeta[ro2,1/4]-Zeta[ro2,3/4]
     f4[ro2]
     g4[8714.2]
     Plot[g4[t],{t,8714.1,8714.4}]

     American Institute of Mathematics, 360 Portage Avenue, Palo Alto, CA 94306
     E-mail address: conrey@aimath.org, xianjin@math.Stanford.EDU

  Current address for Xian-Jin Li: Department of Mathematics, Brigham Young
University, Provo, Utah 84602 USA
