Computing the moment polynomials of the zeta function

TL;DR

This paper introduces a method to efficiently compute the coefficients of moment polynomials related to the Riemann zeta function and uses these computations to test conjectures about zeta moments up to height 10^8.

Contribution

It presents a novel acceleration technique for calculating zeta moment polynomials and applies it to verify conjectures at high heights.

Findings

Computed moment polynomials for k ≤ 13

Experimentally tested conjectures up to height 10^8

Provided numerical evidence supporting conjectured asymptotics

Abstract

We describe a method to accelerate the numerical computation of the coefficients of the polynomials $P_k(x)$ that appear in the conjectured asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out our method to compute the moment polynomials for $k \leq 13$, and used these to experimentally test conjectures for the moments up to height $10^8$.