Moments of zeta and correlations of divisor-sums: IV

TL;DR

This paper advances the understanding of moments of the Riemann zeta-function by analyzing Type II sums using circle method techniques to derive lower order terms in asymptotic formulas for divisor sum correlations.

Contribution

It introduces a new approach to study Type II sums with a convolution of shifted sums, extending the analysis of moments of the zeta-function.

Findings

Derived asymptotic formulas for mean squares of Dirichlet polynomials with divisor coefficients.

Established methods for lower order term calculations in divisor sum correlations.

Extended the analysis to longer Dirichlet polynomials up to length T^3.

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 begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums 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 $T^3$ with divisor functions as coefficients.