The mean square of the product of ζ(s) with Dirichlet polynomials

2014/11/28 by Bettin, Sandro, Chandee, Vorrapan, Radziwill, Maksym
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1411.7764

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 + δ}, with δ= 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.

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