Schoenberg’s Theory of Totally Positive Functions and the Riemann Zeta Function

In a series of papers in the 1950s Schoenberg investigated the properties of totally positive functions [10, 29, 30, 31, 34]. He found several characterizations and used total positivity to prove fundamental properties of splines [34, 33]. 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 $\Lambda$ on $\mathbb{R}$ is totally positive, if for every $n\in\mathbb{N}$ and every two sets of increasing numbers $x_{1}<x_{2}<\dots<x_{n}$ and $y_{1}<y_{2}<\dots<y_{n}$ the matrix $\Big{(}\Lambda(x_{j}-y_{k})\Big{)}_{j,k=1,\dots,n}$ has non-negative determinant:

If in addition $\Lambda$ is integrable, then $\Lambda$ is called a Polya frequency function.

If $\Lambda$ is totally positive and not equal to $e^{ax+b}$, there exist an exponential $e^{cx}$, such that $\Lambda_{1}(x)=e^{cx}\Lambda(x)$ is a Polya frequency function, i.e., $\Lambda_{1}$ 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 [15, 13, 14], where they were instrumental in the derivation of optimal results.

The Laguerre-Polya class. An entire function $\Psi$ of order at most $2$ belongs to the Laguerre-Polya class, if its Hadamard factorization is of the form

where $\delta_{j}^{-1}\in\mathbb{R}$ are the zeros of $\Psi$, $m$ is the order of the zero at $0$, $\gamma\geq 0$, $\delta\in\mathbb{R}$, and

Thus the Laguerre-Polya class consists of entire functions of order two with convergence 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 analytic 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 results about totally positive functions were derived by Schoenberg in a series of papers [29, 30, 31, 34]. A comprehensive treatment is contained in Karlin’s monograph [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 characterization of Schoenberg [31].

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 $(0,\infty)\times\mathbb{R}\times\ell^{2}(\mathbb{Z})$.

By using the Fourier transform instead of the Laplace transform, Schoenberg’s theorem can be recast as follows: A function $\Lambda$ is totally positive and integrable, if and only if its Fourier transform possesses the factorization

where $C>0$, $\gamma\geq 0$, $\delta,\delta_{j}\in\mathbb{R}$ and $\sum_{j=1}^{\infty}\delta_{j}^{2}<\infty$ (and the product in (5) may also be finite).

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 $\Lambda$ converges in some vertical strip $\{z\in\mathbb{C}:\alpha<\mathrm{Re}\,z<\beta\}$ that does not contain $0$.

A similar result holds for one-sided totally positive functions [31, Thm. 2].

Elementary examples. If $\hat{\Lambda}(\tau)=(1+2\pi i\delta\tau)^{-1}$, then $\Lambda(x)=\delta^{-1}e^{-x/\delta}\chi_{[0,\infty)}(x)$ is the one-sided exponential function. For $\hat{\Lambda}(\tau)=e^{-\pi\gamma\tau^{2}}$ for $\gamma>0$, we obtain the Gaussian $\Lambda(x)=\gamma^{-1/2}e^{-\pi x^{2}/\gamma}$. 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 $\Lambda=\Lambda_{1}\ast\Lambda_{2}$ of two Polya frequency functions $\Lambda_{1},\Lambda_{2}$ 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 $\Psi$ with $\Psi(0)>0$ and $\Psi\neq e^{as+b}$ is in the Laguerre-Polya class, if and only if the Fourier transform of $1/\Psi$ is a Polya frequency function.

The Riemann hypothesis and totally positive functions. Let $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ for $s\in\mathbb{C},\mathrm{Re}\,s>1,$ be the Riemann zeta function and let

be the Riemann xi-functions (where $\Gamma$ is the usual gamma function). Then the functional equation for the Riemann zeta function is expressed by the symmetry

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 [18, 17, 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 $\Xi$ has only real zeros, in other words, $\Xi$ 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.

Let us make this statement a bit more explicit by taking the Fourier transform instead of the Laplace transform.

The growth of $\xi$ in the complex plane is [35] $|\xi(s)|=\mathcal{O}\Big{(}e^{A|s|\ln|s|}\Big{)}\,,$ and on the positive real line

Consequently $\frac{1}{\xi(\sigma)}\leq Ce^{-|\sigma|\log|\sigma|/2}$ decays super-exponentially. Since $\zeta$ and thus $\xi$ do not have any real zeros in the interval $[0,1]$ and $\zeta>0$ on $(1,\infty)$, the function $\xi$ is therefore strictly positive on $\mathbb{R}$ and $1/\xi$ is integrable. Thus its Fourier transform is well-defined.

Using $s=2\pi i\tau$, we can rewrite (13) as a Fourier transform. The inversion formula for the Fourier transform now yields

which is (12).

Using the symmetry of $\Xi$, there is an alternative formulation of Theorem 3 with the restricted Laguerre-Polya class defined in (7). Since $\Xi$ is symmetric, it can be written as $\Xi(s)=\Xi_{1}(-s^{2})$ for an entire function $\Xi_{1}$ of order $1/2$. Furthermore, $\Xi$ has only real zeros, if and only if $\Xi_{1}$ has only negative zeros (with convergence exponent at most $1$). The characterization of one-sided Polya frequency functions yields the following equivalence.

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 $\zeta$ and to the well-known criteria for the Riemann hypothesis. Theorem 4 requires only the values of $\zeta$ on the real line to probe the secrets of $\zeta$ in the critical strip. This fact is remarkable, but the price to pay is the added difficulty to extract any meaningful statements about $\xi$ on the critical strip from its restriction to $\mathbb{R}$. This seems much harder, if not impossible.

To work with Theorem 4, one would need a viable expression for the Fourier-Laplace transform of $1/\xi$, but there seems to be none. The $1$-positivity in (1) says that $\Lambda\geq 0$, which is equivalent to the Fourier transform $\hat{\Lambda}=1/\xi$ to be positive definite by Bochner’s theorem. Explicitly, we would need to know that, for all choices of $c_{j}\in\mathbb{C},\tau_{j}\in\mathbb{R},j=1,\dots,n$, and all $n\in\mathbb{N}$, we have $\sum_{j,k=1}^{n}c_{j}\overline{c_{k}}\xi(\tfrac{1}{2}+\tau_{j}-\tau_{k})^{-1}\geq 0$. Not even this property of $1/\xi$ seems to be known. It is therefore unlikely that much is gained by Theorems 3 – 5.

By contrast, the Fourier transform of $\Xi(x)$ 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 $\{0,-1,-2,\dots\}$ with multiplicity one yields the entire function

where $\gamma$ is the Euler constant. By a classical result $\Psi$ is the reciprocal of the $\Gamma$-function $\Gamma(s)=\int_{0}^{\infty}x^{s-1}e^{-x}\,dx$. Consequently, the Laplace transform of $\Psi(s)^{-1}=\Gamma(s)$ is a totally positive function. Indeed, using the substitution $x=e^{-t}$ in the definition of $\Gamma$, one obtains

Theorem 1 implies that

is totally positive. By removing the pole of $\Gamma$ at $0$, we obtain

Consequently $\Lambda_{1}(x)=e^{-x-e^{-x}}$ is a Polya frequency function.

(ii) The zero set $\mathbb{Z}$ with simple zeros yields $\Psi(s)=\frac{\sin\pi s}{\pi}$. By Theorem 1, $1/\Psi$ is the Laplace transform of a totally positive function on a suitable strip of convergence. Schoenberg’s calculation yields the totally positive function

(iii) Finally the zero set $\{-n^{2}:n\in\mathbb{N}\}$ yields the entire function

The associated totally positive function is the Jacobi theta function

All three functions show up prominently in the treatment of the functional equation of the zeta function: $\Gamma$ 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 property of Polya frequency functions is their smoothing property or variation diminishing 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 $\mathbb{R}$ is measured either by the number of sign changes or by the number of real zeros. Formally, given $f:\mathbb{R}\to\mathbb{R}$ let

and let $N(f)$ be the number of real zeros of $f$ counted with multiplicity.

Given a function $\Lambda$, let $T_{\Lambda}$ be the convolution operator $T_{\Lambda}f=f\ast\Lambda$. Schoenberg’s second characterization of Polya frequency functions is as follows [30].

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 obtains the following consequence.

Intrinsic characterizations of the Laguerre-Polya class. There are several characterizations of the Laguerre-Polya class that require only their properties as entire 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 $F(s)\sim\sum_{j=0}^{\infty}a_{j}s^{j}$ yields a differential operator $F(D)p(x)=\sum_{j=0}^{\infty}a_{j}D^{j}p(x)$ with $D=\frac{d}{dx}$. The differential operator is well-defined at least on polynomials, and the mapping $F\mapsto F(D)$ is an algebra homomorphism and thus provides a simple functional calculus.

Applying condition (iv) to the polynomials $x^{n-1}(1+x)^{2}$, one obtains a necessary condition on the Taylor coefficients of a function in the Laguerre-Polya class, namely the so-called Turan inequalities.

Applying condition (v) to polynomials of the form $p(x)=(\sum_{k=1}^{n}a_{k}x^{k})^{2}$ and working out $\Psi(D)^{-1}p$, one obtains the following necessary condition for the Laguerre-Polya class [24, p. 235].

However, the positivity of the Hankel matrices is not sufficient for $\Psi$ 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 $\Xi$ yields an important equivalence of the Riemann hypothesis. The polynomials in condition (iii) are nowadays called Jensen polynomials. In modern language (iii) says that “the Jensen polynomials for the Riemann function $\Xi(s)$ must be hyperbolic”. Significant recent progress on this equivalence is reported in [12].

The relations between the Jensen polynomials, the multiplier sequences of condition (iv), and the Turan inequalities and their generalizations have been studied in depth by Craven, Csordas, and Varga [7, 6, 9] who found many additional equivalences to the Riemann hypothesis. A particular highlight is their proof that $\Xi$, or rather the Taylor coefficients of $\Xi(\sqrt{s})$ 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 $(a_{n})$ is a Polya frequency sequence [1]. This means that the infinite upper triangular Toeplitz matrix $A$ with entries $A_{jk}=a_{k-j}$, if $k\geq j$ and $A_{jk}=0$, 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 $\Lambda$ is assumed to be a Polya frequency function, $\Lambda$ must decay exponentially [31, Lemma 2], therefore its moments of all orders exist. Let

be the $n$-th moment. By expanding the exponential $e^{-sx}=\sum_{j=1}^{\infty}\frac{(-s)^{j}}{j!}x^{j}$ we express the Laplace transform of $\Lambda$ as a power series

Since $\Lambda\not\equiv 0$ and $\Lambda\geq 0$, we have $F(0)>0$, and its reciprocal also possesses a power series expansion around $0$ with a positive radius of convergence

Next, we consider the convolution of $\Lambda$ with a polynomial $p$ of degree $N$ and relate it to the moments of $\Lambda$:

By Corollary 7 the number of real zeros of $q$ (counted with multiplicity) does not exceed the number of real zeros of $p$,

Using the functional calculus, we can invert $F(D)$ and recover $p$ from $q=\Lambda\ast p$ via

For the monomial $q(x)=x^{n}$ we obtain the polynomial

of degree $n$. Since $x^{n}=F(D)q_{n}$, (18) implies the count of zeros (with multiplicities)

For every $n$, $q_{n}$ therefore has only real zeros. This is precisely condition (iii) of Theorem 8, and we conclude that $\Psi$ 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.