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arXiv · 1411.7764

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

Sandro Bettin·Vorrapan Chandee·Maksym Radziwill
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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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Sandro Bettin, Vorrapan Chandee, Maksym Radziwill. 2014-11-28. The mean square of the product of $ζ(s)$ with Dirichlet polynomials. https://arxiv.org/abs/1411.7764

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