[Paper Review] Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis

André Voros|arXiv (Cornell University)|Mar 18, 2014

Analytic Number Theory Research17 references4 citations

TL;DR

This paper introduces 'superzeta' functions built from the nontrivial zeros of the Riemann zeta function, using them to derive an exponential-asymptotic criterion for the Riemann Hypothesis (RH). By analyzing the large-order behavior of Keiper–Li coefficients via the method of steepest descents, the authors show that RH holds if and only if a novel sequence of central expansion coefficients remains positive, providing a sharp, computable criterion based solely on asymptotic growth of these coefficients.

ABSTRACT

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper--Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann Hypothesis that only refers to the large-order Keiper--Li coefficients. As a new result, that criterion, then Li's criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for Keiper--Li).

Motivation & Objective

• To develop generalized zeta functions—'superzeta' functions—based on the nontrivial zeros of the Riemann zeta function, extending the properties of the Hurwitz zeta function.
• To apply these superzeta functions to the Keiper–Li coefficients, enabling asymptotic analysis of their large-order behavior.
• To establish a new, sharp exponential-asymptotic criterion for the Riemann Hypothesis based on the growth and oscillatory behavior of these coefficients.
• To transpose Li’s criterion to a new sequence of coefficients derived from a Taylor expansion centered at s = 1/2 rather than s = 1.
• To provide a computable, analytic criterion for RH that depends only on the asymptotic behavior of a single sequence of coefficients.

Proposed method

• The paper constructs superzeta functions as symmetric functions of the nontrivial zeros of the Riemann zeta function, defined via zeta-regularized products and Mellin transform representations.
• It employs the method of steepest descents to analyze the large-order asymptotics of the Keiper–Li coefficients, identifying the dominant saddle point as a function of the truth of the Riemann Hypothesis.
• The authors derive a new sequence of coefficients, denoted $\lambda_n^0$, based on a Taylor expansion of the Riemann xi function around $s = 1/2$, analogous to the Keiper–Li coefficients but centered at the critical line.
• The functional relation and analytic continuation of the superzeta functions are established using Hankel-type integrals and Mellin transform techniques, mirroring the structure of the Hurwitz zeta function.
• The paper uses residue calculus and contour integration to express the central coefficients $\lambda_n^0$ as contour integrals involving $\log \Xi_0(1/2 + t)$, enabling asymptotic analysis.
• It applies the Hadamard product formula for the xi function to relate the coefficients $\lambda_n^0$ to the zeros $\rho_k$, revealing their sign dependence on the location of the zeros relative to the critical line.

Experimental results

Research questions

• RQ1 Does the large-order asymptotic behavior of the Keiper–Li coefficients exhibit a distinct signature if the Riemann Hypothesis is true or false?
• RQ2 Can a new sequence of coefficients, derived from a Taylor expansion at $s = 1/2$, serve as a valid alternative to the Keiper–Li coefficients for testing the Riemann Hypothesis?
• RQ3 What is the precise exponential-asymptotic criterion for the Riemann Hypothesis based on the growth and oscillatory nature of the coefficients $\lambda_n^0$?
• RQ4 How do the superzeta functions over the Riemann zeros inherit and generalize the analytic and functional properties of the Hurwitz zeta function?
• RQ5 Can the method of steepest descents be effectively applied to extract the dominant asymptotic contribution to the Keiper–Li coefficients, with the saddle point's behavior directly signaling the truth of the Riemann Hypothesis?

Key findings

• If the Riemann Hypothesis is false, the central coefficients $\lambda_n^0$ exhibit exponentially growing oscillatory behavior due to complex zeros off the critical line, specifically $\lambda_n^0 \sim -\sum_{\{\arg\tau_k > 0\}} y_{k,-}^{-n} + \text{c.c.}$, where $|y_{k,-}^{-1}| > 1$.
• If the Riemann Hypothesis is true, the coefficients $\lambda_n^0$ grow asymptotically as $\frac{1}{2}n(\log n - 1 + \gamma - \log 2\pi)$, matching the known asymptotic form of the Keiper–Li coefficients.
• The Riemann Hypothesis is equivalent to the positivity of all $\lambda_n^0$ for $n \geq 1$, providing a new, sharp criterion that only depends on the sign of these coefficients.
• The central coefficients $\lambda_n^0$ are explicitly given by $\lambda_n^0 = \sum_k (2 - y_{k,-}^{-n} - y_{k,+}^{-n})$, where $y_{k,\pm} = \exp(\pm i\theta_k)$, and are real and positive if and only if all $\rho_k$ lie on the critical line.
• The paper derives multiple equivalent integral representations for $\lambda_n^0$, including a contour integral in the complex $y$-plane and a higher-order derivative formula involving $\log \Xi_0(1/2 + t)$, enabling rigorous asymptotic analysis.
• The superzeta function $\mathcal{Z}_0(s)$, defined over the zeros of the Riemann xi function, is shown to have a meromorphic continuation and explicit special-value formulae, mirroring the properties of the Hurwitz zeta function.