Analysis of Riemann Zeta-Function Zeros using Pochhammer Polynomial Expansions

TL;DR

This paper investigates the zeros of the Riemann Xi-function through Pochhammer polynomial expansions, revealing a growth pattern of minimal beta-sequences that supports the Riemann Hypothesis.

Contribution

It introduces a novel approach using Pochhammer polynomial expansions and minimal beta-sequences to analyze the zeros of the Riemann Xi-function, providing new insights into the hypothesis.

Findings

Minimal beta(n) sequences grow sub-logarithmically

Approximants converge to Xi(t) when beta(n)=o(log(n))

Analysis supports the validity of the Riemann Hypothesis

Abstract

The Riemann Xi-function Xi(t) belongs to a family of entire functions which can be expanded in a uniformly convergent series of symmetrized Pochhammer polynomials depending on a real scaling parameter beta. It can be shown that the polynomial approximant Xi(n,t,beta) to Xi(t) has distinct real roots only in the asymptotic scaling limit beta->infinity. One may therefore infer the existence of increasing beta-sequences beta(n)->infinity for n->infinity, such that Xi(n,t,beta(n)) has real roots only for all n, and to each entire function it is possible to associate a unique minimal beta-sequence fulfilling a specific difference equation. Numerical analysis indicates that the minimal beta(n) sequence associated with the Riemann Xi(t) has a distinct sub-logarithmic growth rate, and it can be shown that the approximant Xi(n,t,beta(n)) converges to Xi(t) when n->infinity if beta(n)=o(log (n)). Invoking the Hurwitz theorem of complex analysis, and applying a formal analysis of the asymptotic properties of minimal beta-sequences, a fundamental mechanism is identified which provides a compelling confirmation of the validity of the Riemann Hypothesis.