Mean values of long Dirichlet polynomials with divisor coefficients

TL;DR

This paper derives an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients, including lower order terms and a power-saving error, confirming a conjecture for divisor functions.

Contribution

It provides a stronger form of a conjecture by Conrey and Gonek for divisor functions, extending previous results on mean values of Dirichlet polynomials.

Findings

Asymptotic formula with main and lower order terms

Power saving error term established

Confirmation of a conjecture for divisor functions

Abstract

In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is derived from a more general theorem on mean values of long smoothed Dirichlet polynomials that was previously established by the second and third authors. We thus establish a stronger form of a conjecture of Conrey and Gonek in the case of divisor functions.