                                                   SCHOENBERG’S THEORY OF TOTALLY POSITIVE
                                                  FUNCTIONS AND THE RIEMANN ZETA FUNCTION

                                                                            KARLHEINZ GRÖCHENIG

                                                  Abstract. We review Schoenberg’s characterization of totally positive func-




arXiv:2007.12889v1 [math.NT] 25 Jul 2020
                                                  tions and its connection to the Laguerre-Polya class. This characterization yields
                                                  a new condition that is equivalent to the truth of the Riemann hypothesis.


                                             In a series of papers in the 1950s Schoenberg investigated the properties of totally
                                           positive functions [10, 29–31, 34]. He found several characterizations and used total
                                           positivity to prove fundamental properties of splines [33, 34]. The purpose of this
                                           note is to survey some aspects of Schoenberg’s work on totally positive functions,
                                           to advertize the connection between totally positive functions and the Riemann
                                           hypothesis, and to provide some mathematical entertainment.
                                             One may speculate whether Schoenberg himself thought about the Riemann zeta
                                           function. He was the son-in-law of the eminent number theorist Edmund Landau,
                                           he collaborated with Polya, he knew deeply the work of Polya and Schur about
                                           the Laguerre-Polya class of entire functions that remains influential in the study of
                                           the Riemann hypothesis. Yet, to my knowledge, he never mentioned any number
                                           theory in his work on totally positive functions and splines; by the same token,
                                           Schoenberg’s name is not mentioned in analytic number theory.
                                              Totally positive functions. A measurable function Λ on R is totally positive, if
                                           for every n ∈ N and every twosets of increasing
                                                                                           numbers x1 < x2 < · · · < xn and
                                           y1 < y2 < · · · < yn the matrix Λ(xj − yk )        has non-negative determinant:
                                                                                               j,k=1,...,n
                                                                                           
                                           (1)                           det(Λ(xj − yk )                     ≥ 0.
                                                                                              j,k=1,...,n

                                           If in addition Λ is integrable, then Λ is called a Polya frequency function.
                                              If Λ is totally positive and not equal to eax+b , there exist an exponential ecx ,
                                           such that Λ1 (x) = ecx Λ(x) is a Polya frequency function, i.e., Λ1 is totally positive
                                           and integrable [31, Lemma 4]. It is usually no loss of generality to restrict to Polya
                                           frequency functions.
                                              The class of totally positive functions played and plays an important role in
                                           approximation theory, in particular in spline theory [34], and in statistics [11, 19].
                                           In a different and rather surprising direction, totally positive functions appear in the
                                           representation theory of infinite dimensional motion groups [23]. Recently, totally
                                           positive functions appeared in sampling theory and in time-frequency analysis [13–
                                           15], where they were instrumental in the derivation of optimal results.

                                             K. G. was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF).
                                                                                          1
2                             KARLHEINZ GRÖCHENIG

  The Laguerre-Polya class. An entire function Ψ of order at most 2 belongs to
the Laguerre-Polya class, if its Hadamard factorization is of the form
                                       ∞
                                       Y
                            m −γs2 +δs
(2)             Ψ(s) = Cs e              (1 + δj s)e−δj s  s ∈ C,
                                       j=1

where δj−1 ∈ R are the zeros of Ψ, m is the order of the zero at 0, γ ≥ 0, δ ∈ R,
and
                                         ∞
                                         X
(3)                             0<γ+           δj2 < ∞ .
                                         j=1

Thus the Laguerre-Polya class consists of entire functions of order two with con-
vergence exponent two with only real zeros. While the study of the distribution of
zeros of entire functions is a perennial topic in complex analysis and of interest in
its own right [21], the Laguerre-Polya class has gained special prominence in ana-
lytic number theory: the Riemann hypothesis says that a relative of the Riemann
zeta function belongs to the Laguerre-Polya class.
  Schoenberg’s characterization of totally positive functions. The fundamental re-
sults about totally positive functions were derived by Schoenberg in a series of
papers [29–31, 34]. A comprehensive treatment is contained in Karlin’s mono-
graph [19, Ch. 7].
  The notions of totally positive functions and Laguerre-Polya class are seemingly
unrelated, yet there is a deep connection between them through the following char-
acterization of Schoenberg [31].
Theorem 1. (i) If Λ is a Polya frequency function, then its (two-sided) Laplace
transform converges in a vertical strip {z ∈ C : α < Re z < β}, α < 0 < β, and
                            Z ∞
                                                   1
(4)                               Λ(x)e−sx dx =
                              −∞                  Ψ(s)
is the reciprocal of a function Ψ in the Laguerre-Polya class with Ψ(0) > 0.
   (ii) Conversely, if Ψ is in the Laguerre-Polya class with Ψ(0) > 0, then its
reciprocal 1/Ψ is the Laplace transform of a Polya frequency function Λ.
   This is a fascinating theorem, because it relates two function classes that seem to
bear absolutely no resemblance to each other. Schoenberg’s theorem establishes a
bijection between the class of Polya frequency functions, the Laguerre-Polya class,
and yields a parametrization by the set (0, ∞) × R × ℓ2 (Z).
   By using the Fourier transform instead of the Laplace transform, Schoenberg’s
theorem can be recast as follows: A function Λ is totally positive and integrable,
if and only if its Fourier transform possesses the factorization
                                           ∞
                                           Y
                                   2
(5)                  Λ̂(τ ) = Ce−γτ +2πiδτ   (1 + 2πiδj τ )−1 e2πiδj τ
                                         j=1
                                    P∞      2
where C > 0, γ ≥ 0, δ, δj ∈ R and      j=1 δj < ∞ (and the product in (5) may also
be finite).
             TOTALLY POSITIVE FUNCTIONS AND THE ZETA FUNCTION                         3

  If we drop the condition of integrability and exclude exponential functions,
then the representation (5) still holds for every totally positive function, but their
Laplace transform of Λ converges in some vertical strip {z ∈ C : α < Re z < β}
that does not contain 0.
  A similar result holds for one-sided totally positive functions [31, Thm. 2].

Theorem 2. (i) If Λ is a Polya frequency function with support in [0, ∞), then its
Laplace transform converges in a half-plane {z ∈ C : −α < Re z}, α > 0, and
                            Z ∞
                                                 1
(6)                              Λ(x)e−sx dx =
                              0                 Ψ(s)
is the reciprocal of an entire function Ψ with Hadamard factorization
                                              ∞
                                              Y
                                         δs
(7)                           Ψ(s) = Ce             (1 + δj s) ,
                                              j=1
                   P
with δ ∈ R, δj ≥ 0, δj < ∞.
  (ii) Conversely, if Ψ possesses the factorization (7), then its reciprocal 1/Ψ is
the Laplace transform of a Polya frequency function Λ with support in [0, ∞).

   Elementary examples. If Λ̂(τ ) = (1 + 2πiδτ )−1 , then Λ(x) = δ −1 e−x/δ χ[0,∞) (x)
                                                            2
is the one-sided exponential function. For Λ̂(τ ) = e−πγτ for γ > 0, we obtain the
                            2
Gaussian Λ(x) = γ −1/2 e−πx /γ . In both cases, it is easy to check directly that these
functions are totally positive.
   The proof of the implication (ii) of Theorem 1 is based on the (non-trivial) fact
that the convolution Λ = Λ1 ∗ Λ2 of two Polya frequency functions Λ1 , Λ2 is again
a Polya frequency function. The converse in Theorem 1 lies much deeper, and
Schoenberg used heavily several results of Polya about functions in the Laguerre-
Polya class [24, 27]. See the end of this note for the essential step of the argument.
   Schoenberg’s motivation was the characterization and deeper understanding of
totally positive functions, and thus the implication (i) and the factorization (5) can
be considered his main insight about totally positive functions. However, instead
of reading Schoenberg’s theorem as a characterization of totally positive functions,
one may read it as a characterization of the Laguerre-Polya class. A function Ψ
with Ψ(0) > 0 and Ψ 6= eas+b is in the Laguerre-Polya class, if and only if the
Fourier transform of 1/Ψ is a Polya frequency function.
   The Riemann hypothesis and totally positive functions. Let ζ(s) = ∞           −s
                                                                          P
                                                                            n=1 n   for
s ∈ C, Re s > 1, be the Riemann zeta function and let
                                                      s
(8)                        ξ(s) = 21 s(s − 1)π −s/2 Γ     ζ(s)
                                                       2
                                    1        
(9)                        Ξ(s) = ξ      + is
                                      2
4                              KARLHEINZ GRÖCHENIG

be the Riemann xi-functions (where Γ is the usual gamma function). Then the
functional equation for the Riemann zeta function is expressed by the symmetry
(10)                ξ(s) = ξ(1 − s)       and       Ξ(s) = Ξ(−s)
for the xi-functions.
   The Riemann hypothesis conjectures that all non-trivial zeros of the zeta function
lie on the critical line 1/2 + it. See the monographs [17, 18, 35], the two volumes
about equivalents of the Riemann hypothesis [3, 4] or the survey articles [2, 5].
   Expressed in terms of the xi-functions, the Riemann hypothesis states that Ξ
has only real zeros, in other words, Ξ belongs to the Laguerre-Polya class. Thus
many investigations of the zeta function involve complex analysis related to the
Laguerre-Polya class. Schoenberg’s theorem immediately leads to the following
equivalent condition for the Riemann hypothesis to hold.
Theorem 3. The Riemann hypothesis holds, if and only if there exists a Polya
frequency function Λ, such that
                       Z ∞
                 1
(11)                 =      Λ(x)e−sx dx for s ∈ C, |Re s| < t0 ,
                Ξ(s)    −∞

where 1/2 + it0 is the first zero of the zeta function on the critical line.
   Let us make this statement a bit more explicit by taking the Fourier transform
instead of the Laplace transform.
Theorem 4. The Riemann hypothesis holds, if and only if
                              Z ∞
                            1           1
(12)                Λ(x) =            1      e−ixτ dτ
                           2π −∞ ξ( 2 + τ )
is a Polya frequency function.
                                                                               
                                                                  A|s| ln |s|
  The growth of ξ in the complex plane is [35] |ξ(s)| = O e                         , and on the
positive real line
                         ln ξ(σ) ≍ 12 σ log σ   σ > 1.
                  1
Consequently ξ(σ)    ≤ Ce−|σ| log |σ|/2 decays super-exponentially. Since ζ and thus ξ
do not have any real zeros in the interval [0, 1] and ζ > 0 on (1, ∞), the function ξ
is therefore strictly positive on R and 1/ξ is integrable. Thus its Fourier transform
is well-defined.
   Using s = 2πiτ , we can rewrite (13) as a Fourier transform. The inversion
formula for the Fourier transform now yields
                                    Z ∞
                                             1
                          Λ(x) =                  e2πixτ dτ
                                         Ξ(2πiτ )
                                    Z−∞∞
                                               1
                                =                        e2πixτ dτ ,
                                      −∞ ξ(1/2 −  2πτ  )
which is (12).
              TOTALLY POSITIVE FUNCTIONS AND THE ZETA FUNCTION                         5

   Using the symmetry of Ξ, there is an alternative formulation of Theorem 3 with
the restricted Laguerre-Polya class defined in (7). Since Ξ is symmetric, it can be
written as Ξ(s) = Ξ1 (−s2 ) for an entire function Ξ1 of order 1/2. Furthermore,
Ξ has only real zeros, if and only if Ξ1 has only negative zeros (with convergence
exponent at most 1). The characterization of one-sided Polya frequency functions
yields the following equivalence.
Theorem 5. The Riemann hypothesis holds, if and only if there exists a Polya
frequency function Λ with support in [0, ∞), such that
                         Z ∞
                  1
(13)                   =     Λ(x)e−sx dx      for s ∈ C, Re s > α ,
                Ξ1 (s)    0

for some α < 0.
   These equivalences seem to be new. Schoenberg’s name is not even mentioned
in [3, 4] on equivalents of the Riemann hypothesis.
   It is interesting that the characterization of Theorem 4 is “orthogonal” to most
research on ζ and to the well-known criteria for the Riemann hypothesis. Theorem 4
requires only the values of ζ on the real line to probe the secrets of ζ in the critical
strip. This fact is remarkable, but the price to pay is the added difficulty to extract
any meaningful statements about ξ on the critical strip from its restriction to R.
This seems much harder, if not impossible.
   To work with Theorem 4, one would need a viable expression for the Fourier-
Laplace transform of 1/ξ, but there seems to be none. The 1-positivity in (1) says
that Λ ≥ 0, which is equivalent to the Fourier transform Λ̂ = 1/ξ to be positive
definite by Bochner’s theorem. Explicitly, we would need to know     P that, for all
choices of cj ∈ C, τj ∈ R, j = 1, . . . , n, and all n ∈ N, we have nj,k=1 cj ck ξ( 12 +
τj − τk )−1 ≥ 0. Not even this property of 1/ξ seems to be known. It is therefore
unlikely that much is gained by Theorems 3 – 5.
   By contrast, the Fourier transform of Ξ(x) on the critical line (!) was already
known to Riemann (see [35, 2.16.1]) and is the starting point of a program to prove
the Riemann hypothesis that goes back to Polya [25]. After important work of de
Bruijn, Hejhal, and Newman this line of thought has recently culminated in the
resolution of the Newman conjecture by Rodgers and Tao [28].
  Some non-trivial Polya frequency functions. Perhaps Schoenberg had also the
Riemann hypothesis in mind, when he investigated Polya frequency functions. The
examples in [29, 31] of totally positive functions smell of the zeta function.
  (i) The zero set {0, −1, −2, . . . } with multiplicity one yields the entire function
                                          ∞
                                     γs
                                          Y          s −s/n
(14)                        Ψ(s) = e s        (1 +     )e   ,
                                          n=1
                                                     n

where γ is the Euler   constant. By a classical result Ψ is the reciprocal of the Γ-
                 R ∞ s−1
function Γ(s) = 0 x e−x dx. Consequently, the Laplace transform of Ψ(s)−1 =
Γ(s) is a totally positive function. Indeed, using the substitution x = e−t in the
6                               KARLHEINZ GRÖCHENIG

definition of Γ, one obtains
                                Z ∞
                                       e−e e−sx dx
                                           −x
(15)                   Γ(s) =                            Re s > 0 .
                                  −∞

Theorem 1 implies that
                                   Λ(x) = e−e
                                                   −x



is totally positive. By removing the pole of Γ at 0, we obtain
                  Z ∞                Z ∞
                        ′   −sx
                                          e−x e−e e−sx dx ,
                                                 −x
         sΓ(s) =      Λ (x)e dx =                              Res > −1 .
                  −∞                        −∞
                         −x−e−x
Consequently Λ1 (x) = e         is a Polya frequency function.
  (ii) The zero set Z with simple zeros yields Ψ(s) = sinππs . By Theorem 1, 1/Ψ is
the Laplace transform of a totally positive function on a suitable strip of conver-
gence. Schoenberg’s calculation yields the totally positive function
                                              1
                                  Λ(x) =          .
                                          1 + e−x
  (iii) Finally the zero set {−n2 : n ∈ N} yields the entire function
                                ∞
                                Y          s        1√          √
                    Ψ(s) = s        (1 +      ) = −    −s sin π   −s .
                                n=1
                                           n2       π
The associated totally positive function is the Jacobi theta function
                             (P
                                 ∞          j −j 2 x
                                 j=−∞ (−1) e         for x > 0
                    Λ(x) =
                               0                     for x ≤ 0 .
All three functions show up prominently in the treatment of the functional equation
of the zeta function: Γ is contained in the definition of the xi-function, sin in the
formulation of the functional equation, and a Jacobi theta function is used in several
proofs of the functional equation (Riemann’s original proof, see [35]).
   Intrinsic characterization of Polya frequency functions. The fundamental prop-
erty of Polya frequency functions is their smoothing property or variation dimin-
ishing property. The relevance of smoothing properties for many applications is
outlined in Schoenberg’s survey [32]. In this context the variation of a real-valued
function on R is measured either by the number of sign changes or by the number
of real zeros. Formally, given f : R → R let
(16)
    v(f ) = max #{n ∈ N : ∃xj ∈ R, x0 < x1 < · · · < xn with f (xj )f (xj+1 ) < 0} ,
and let N(f ) be the number of real zeros of f counted with multiplicity.
  Given a function Λ, let TΛ be the convolution operator TΛ f = f ∗Λ. Schoenberg’s
second characterization of Polya frequency functions is as follows [30].
Theorem 6. Let Λ be integrable and continuous. Then Λ is variation diminishing,
i.e.,
                               v(TΛ f ) ≤ v(f )
              TOTALLY POSITIVE FUNCTIONS AND THE ZETA FUNCTION                           7

for all functions that are locally Riemann integrable, if and only if either Λ or −Λ
is a Polya frequency function.
   This characterization is “intrinsic” in the sense that it uses only the properties
of the matrices occurring in the definition (1) of total positivity.
   With a perturbation argument one can replace sign changes with zeros and ob-
tains the following consequence.
Corollary 7. Let Λ be a Polya frequency function. Then for every real-valued
polynomial p the convolution TΛ is zero-decreasing, i.e.,
                                   N(TΛ p) ≤ N(p) .

   Intrinsic characterizations of the Laguerre-Polya class. There are several char-
acterizations of the Laguerre-Polya class that require only their properties as en-
tire functions. This is part of classical complex analysis and the results are due to
Polya and Schur [24,27] building on work of Laguerre, Hadamard, and many others.
These results relate the properties of the zero set to properties of the power series
expansion of an entire function. Before formulating           a sequence of equivalences, we
note that every formal power series F (s) ∼ ∞                  j
                                                   P
                                                      j=0 ja s   yields a differential operator
               P∞         j                    d
F (D)p(x) = j=0 aj D p(x) with D = dx . The differential operator is well-defined
at least on polynomials, and the mapping F 7→ F (D) is an algebra homomorphism
and thus provides a simple functional calculus.
                                        βj j
Theorem 8. Let Ψ(s) = ∞
                                P
                                   j=0 j! s be an entire function. Then the following are
equivalent:
   (i) Ψ belongs to the Laguerre-Polya class.
   (ii) Ψ can be approximated uniformly on compact sets by polynomials with only
real zeros.
   (iii) For all n ∈ N the polynomials pn (x) = nj=0 βj nj xj and qn (x) = nj=0 βj nj xn−j
                                                    P                              P         

have only real zeros.
   (iv) If p(x) = m            j
                   P
                      j=0 c j x is a polynomial with only real, non-positive zeros, then
                               βj cj xj has only P
                          P
the polynomial q(x) =                            real zeros.
                                                       γj j
   If, in addition, Ψ(0) > 0 and Ψ(s) = ∞  1
                                                   j=0 j! s , then the following property is
equivalent to (i) – (iv).
                                     1
   (v) The transform p 7→ Ψ(D)          p is zero-decreasing, i.e., the polynomial q(x) =
  1
              P  ∞ γj (j)
Ψ(D)
      p(x) = j=0 j! p (x) has at most as many real zeros as p (real-valued):
                                         1      
                                      N         p ≤ N(p) .
                                          Ψ(D)
  Applying condition (iv) to the polynomials xn−1 (1 + x)2 , one obtains a neces-
sary condition on the Taylor coefficients of a function in the Laguerre-Polya class,
namely the so-called Turan inequalities.
                               βj j
Corollary 9. If Ψ(s) = ∞
                        P
                           j=0 j! s belongs to the Laguerre-Polya class, then

                        βn2 − βn−1 βn+1 ≥ 0        for all n ∈ N
8                              KARLHEINZ GRÖCHENIG

  Applying condition (v) to polynomials of the form p(x) = ( nk=1 ak xk )2 and
                                                                  P
working out Ψ(D)−1 p, one obtains the following necessary condition for the Laguerre-
Polya class [24, p. 235].
Corollary 10. Assume that Ψ belongs to the Laguerre-Polya
                                                      P∞ γj jclass, Ψ(0) > 0,
          as+b                                   1
Ψ(s) 6= e      and 1/Ψ has the Taylor expansion Ψ(s) = j=0 j! s . Then for every
                                        
n ∈ N the n × n Hankel matrix γj+k j,k=0,...,n−1 is positive definite (and thus
invertible).
   However, the positivity of the Hankel matrices is not sufficient for Ψ to be in the
Laguerre-Polya class, as was proved already by Hamburger [16].
   Theorem 8 and its corollaries are all contained in the seminal papers of Polya
and Schur [24, 27] from 1914 and 1915 and have inspired a century of exciting
mathematics. Each of the equivalent conditions in Theorem 8 is a point of departure
for the study of the Riemann hypothesis.
   No list can do justice to all contributions between 1914 and 2020, so let us
mention only a few directions whose origin is in Polya’s work. Further references
and more detailed history can be found in the cited articles.
   Condition (iii) applied to the Riemann function Ξ yields an important equiva-
lence of the Riemann hypothesis. The polynomials in condition (iii) are nowadays
called Jensen polynomials. In modern language (iii) says that “the Jensen poly-
nomials for the Riemann function Ξ(s) must be hyperbolic”. Significant recent
progress on this equivalence is reported in [12].
   The relations between the Jensen polynomials, the multiplier sequences of con-
dition (iv), and the Turan inequalities and their generalizations have been studied
in depth by Craven, Csordas, and Varga [6, 7, 9] who found many additional equiv-
alences to the Riemann hypothesis. √A particular highlight is their proof that Ξ, or
rather the Taylor coefficients of Ξ( s) satisfy the Turan inequalities [8], thereby
resolving a 60 year old conjecture going back to — Polya.
   Finally let us mention that total positivity enters the investigation of the Laguerre-
Polya class in yet another way. A entire function belongs to the restricted Laguerre-
Polya class defined by (7), if and only if the sequence of its Taylor coefficients (an )
is a Polya frequency sequence [1]. This means that the infinite upper triangular
Toeplitz matrix A with entries Ajk = ak−j , if k ≥ j and Ajk = 0, if k < j has only
positive minors. This aspect of total positivity has been used in [20, 22] for the
investigation of the zeta function.
  From total positivity to the Laguerre-Polya class. By comparing the two intrinsic
characterizations in Theorems 6 and 8 one may guess that the respective conditions
on zero diminishing must play the decisive role in the proof of Theorem 1(i). To give
the gist of this argument, we cannot do better than repeat Schoenberg’s beautiful
argument.
  First, since Λ is assumed to be a Polya frequency function, Λ must decay expo-
nentially [31, Lemma 2], therefore its moments of all orders exist. Let
                                      Z
                                 µn =    xn Λ(x) dx
                                         R
               TOTALLY POSITIVE FUNCTIONS AND THE ZETA FUNCTION                       9

                                                                (−s)j j
be the n-th moment. By expanding the exponential e−sx = ∞
                                                          P
                                                             j=1 j! x we express
the Laplace transform of Λ as a power series
                                      ∞
                                         (−1)j
                    Z               X
(17)                     −sx
                        e Λ(x) dx =            µj sj := F (s) .
                      R              j=0
                                           j!

Since Λ 6≡ 0 and Λ ≥ 0, we have F (0) > 0, and its reciprocal also possesses a power
series expansion around 0 with a positive radius of convergence
                                                   ∞
                                            1     X   βj j
                                 Ψ(s) =         =        s .
                                          F (s)   j=0
                                                      j!

Next, we consider the convolution of Λ with a polynomial p of degree N and relate
it to the moments of Λ:
                                                     N
                                                        (−t)j (j) 
                           Z                     Z X
       q(x) = (Λ ∗ p)(x) =     p(x − t)Λ(t) dt =             p (x) Λ(t) dt
                             R                    R j=0
                                                         j!
               n
               X (−1)j
           =                µj p(j) (x) = F (D)p(x) .
               j=0
                       j!

By Corollary 7 the number of real zeros of q (counted with multiplicity) does not
exceed the number of real zeros of p,
(18)                           N(q) = N(F (D)p) ≤ N(p) .
Using the functional calculus, we can invert F (D) and recover p from q = Λ ∗ p via
                                                     ∞
                              1                     X   βj (j)
                     p(x) =       q(x) = Ψ(D)q(x) =        q (x) .
                            F (D)                   j=0
                                                        j!

For the monomial q(x) = xn we obtain the polynomial
                                          n      
                                    n
                                        X        n n−j
                      qn (x) = Ψ(D)x =       βj    x
                                         j=0
                                                 j

of degree n. Since xn = F (D)qn , (18) implies the count of zeros (with multiplicities)
                                 n = N(xn ) ≤ N(qn ) ≤ n .
For every n, qn therefore has only real zeros. This is precisely condition (iii) of
Theorem 8, and we conclude that Ψ is in the Laguerre-Polya class.
   Summary. Schoenberg’s characterization of totally positive functions implies a
condition equivalent to the Riemann hypothesis. The characterization is interesting
in itself because it involves only the values of the Riemann zeta function on the
real axis. To the best of our knowledge, the characterization of the Laguerre-Polya
class by means of totally positive functions has not yet been tested on the Riemann
zeta function.
10                                 KARLHEINZ GRÖCHENIG

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  Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
A-1090 Vienna, Austria
  E-mail address: karlheinz.groechenig@univie.ac.at
