The mean square of the product of $\zeta(s)$ with Dirichlet polynomials

TL;DR

This paper derives asymptotic formulas for the mean-square of the Riemann zeta-function multiplied by Dirichlet polynomials of specific lengths, advancing understanding of moments and related conjectures in analytic number theory.

Contribution

It extends previous results by providing asymptotic formulas for the mean-square of ζ(s) times Dirichlet polynomials of larger lengths and refines bounds for moments, including the third moment.

Findings

Asymptotic formula for mean-square with Dirichlet polynomial length up to T^{1/2 + δ}

Upper bound for the third moment of ζ(s) of correct order

Asymptotic estimates for Dirichlet polynomials of length up to T^{3/4} with special shape

Abstract

Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + \delta}$, with $\delta = 0.01515....$ As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec, obtaining asymptotic estimates in place of bounds. Using the work of Watt, we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to $T^{3/4}$ provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelof Hypothesis.