Moments of zeta and correlations of divisor-sums: III

TL;DR

This paper advances the understanding of the Riemann zeta-function's moments by refining the calculation of lower order terms in mean square asymptotics using divisor correlations and shifted convolution conjectures.

Contribution

It provides a detailed input of the conjectural divisor sum formulas to accurately compute lower order terms in the zeta function moments.

Findings

Derived precise lower order terms in mean square asymptotics.

Enhanced the divisor correlation approach for zeta moments.

Connected shifted convolution conjectures with moment calculations.

Abstract

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients.