[Paper Review] On Hilbert's 8th Problem

Nicholas G. Polson|arXiv (Cornell University)|Aug 7, 2017

Analytic Number Theory Research4 references3 citations

TL;DR

This paper reformulates the Riemann Hypothesis (RH) via a Hadamard factorization of the Riemann xi-function, showing RH holds if and only if the reciprocal xi-function at 1/2 is the Laplace transform of a generalized gamma convolution (GGC). The key result establishes that RH is equivalent to the existence of a GGC random variable $ H_{1/2}^ heta $ such that $ \xi(1/2)/\xi(1/2 + \sqrt{s}) = \mathbb{E}[\exp(-sH_{1/2}^\xi)] $ for $ s > 0 $, linking analytic number theory to stochastic processes.

ABSTRACT

A Hadamard factorization of the Riemann Xi-function is constructed to characterize the zeros of the zeta function.

Motivation & Objective

• To reformulate the Riemann Hypothesis (RH) in terms of probabilistic properties of the Riemann xi-function.
• To establish a connection between the location of non-trivial zeta zeros and the Laplace transform of a generalized gamma convolution (GGC).
• To prove that RH holds if and only if the reciprocal xi-function at $ 1/2 $ is the Laplace transform of a GGC random variable.
• To use Ramanujan's Master Theorem and Frullani's identity to derive integral representations of the xi-function and verify the GGC condition.

Proposed method

• Constructs a Hadamard factorization of the xi-function using its non-trivial zeros $ \rho = 1/2 \pm i\tau $, leading to an infinite product over $ \tau > 0 $.
• Applies Frullani's identity to express the reciprocal xi-function as an exponential of an integral involving a Lévy measure $ \nu_\xi(t) $, linking it to a Laplace transform.
• Uses Thorin’s condition to show that RH is equivalent to $ \xi(1/2)/\xi(1/2 + \sqrt{s}) $ being the Laplace transform of a GGC random variable $ H_{1/2}^\xi $.
• Employs Ramanujan’s Master Theorem to relate the power series expansion of the reciprocal xi-function to its Laplace transform representation.
• Derives the GGC property of $ H_{1/2}^\xi $ via analytic continuation and closure properties of the GGC class under exponential tilting and composition.
• Verifies that the resulting Laplace transform is analytic in the cut plane $ \mathbb{C} \setminus (-\infty, 0] $, ensuring no zeros of $ \xi(1/2 + s) $ exist in $ \text{Re}(s) > 0 $, thus implying RH.

Experimental results

Research questions

• RQ1 Is the Riemann Hypothesis equivalent to the condition that $ \xi(1/2)/\xi(1/2 + \sqrt{s}) $ is the Laplace transform of a generalized gamma convolution (GGC) for $ s > 0 $?
• RQ2 Can the Hadamard factorization of the xi-function be used to construct a stochastic representation of its reciprocal via exponential families and Lévy measures?
• RQ3 Does the analytic continuation of the Laplace transform of a GGC imply the absence of non-trivial zeros of $ \zeta(s) $ off the critical line $ \text{Re}(s) = 1/2 $?
• RQ4 Can Ramanujan’s Master Theorem be applied to the reciprocal xi-function to derive a power series representation consistent with GGC properties?
• RQ5 Is the GGC property preserved under exponential tilting and composition, and does this imply that $ H_{1/2}^\xi $ is a valid GGC under the assumed conditions?

Key findings

• The Riemann Hypothesis holds if and only if $ \xi(1/2)/\xi(1/2 + \sqrt{s}) = \mathbb{E}[\exp(-sH_{1/2}^\xi)] $ for $ s > 0 $, where $ H_{1/2}^\xi $ is a generalized gamma convolution (GGC).
• The reciprocal xi-function is expressed as $ \prod_{\tau>0} \frac{1}{1 + s^2/\tau^2} $, which is shown to be the Laplace transform of a sum of exponential random variables with rates $ \tau^2 $, confirming the GGC structure under RH.
• The GGC property of $ H_{1/2}^\xi $ is preserved under exponential tilting, and the closure of the GGC class under such operations confirms the validity of the stochastic representation.
• Using Ramanujan’s Master Theorem, the paper derives an integral representation of $ \xi(3/2)/\xi(1/2 + \sqrt{1-s}) $ as $ \Gamma(s)^{-1} \int_0^\infty x^{s-1} \sum_{k=0}^\infty \frac{(-x)^k}{k!} \frac{\xi(3/2)}{\xi(1/2 + \sqrt{1+k})} dx $, which equals $ \mathbb{E}[\exp(sH_\star^\xi)] $, confirming the GGC structure for $ H_\star^\xi $.
• The function $ \xi(3/2)/\xi(1/2 + \sqrt{1+s}) $ is shown to be the Laplace transform of a GGC random variable $ H_\star^\xi $, providing a second stochastic characterization of the xi-function.
• The analytic continuation of the Laplace transform from $ s > 0 $ to the cut plane $ \mathbb{C} \setminus (-\infty, 0] $ implies that $ \xi(1/2 + s) $ has no zeros in $ \text{Re}(s) > 0 $, and by symmetry, no zeros in $ \text{Re}(s) < 1/2 $, thus proving RH.