Jensen Polynomials for the Riemann Xi Function

TL;DR

This paper explores the connection between Jensen polynomials derived from the Riemann xi function and the Riemann Hypothesis, providing effective criteria and analyzing how zeros of derivatives influence polynomial hyperbolicity.

Contribution

It makes the hyperbolicity of Jensen polynomials effective and links the zeros of derivatives of xi to polynomial hyperbolicity, advancing understanding of RH equivalences.

Findings

Hyperbolicity of Jensen polynomials can be effectively determined.

Zeros of xi derivatives influence polynomial hyperbolicity.

Provides new criteria for testing the Riemann Hypothesis.

Abstract

We investigate Riemann's xi function $\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ (here $\zeta(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $\xi(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. P\'olya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials $J^{d,n}(X)$ constructed from certain Taylor coefficients of $\xi(s)$. For each $d\geq 1$, recent work proves that $J^{d,n}(X)$ is hyperbolic for sufficiently large $n$. Here we make this result effective. Moreover, we show how the low-lying zeros of the derivatives $\xi^{(n)}(s)$ influence the hyperbolicity of $J^{d,n}(X)$.