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Published August 2017 | Accepted Version
Journal Article Open

The mean square of the product of the Riemann zeta-function with Dirichlet polynomials


Improving earlier work of Balasubramanian, Conrey and Heath-Brown [1], 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 [8], obtaining asymptotic estimates in place of bounds. Using the work of Watt [19], 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 Lindelöf Hypothesis.

Additional Information

© 2017 by Walter de Gruyter GmbH. Received: 2013-08-07. Revised: 2014-11-07. Published Online: 2015-02-05. The third author was partially supported by NSF grant DMS-1128155. We are very grateful to Brian Conrey for suggesting to us the problem of breaking the 1/2 barrier in Theorem 1 and to Micah B. Milinovich and Nathan Ng for pointing out the paper of Duke, Friedlander, Iwaniec [DFI97a]. We also wish to thank the referee for a very careful reading of the paper and for indicating several inaccuracies and mistakes.

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Accepted Version - 1411.7764.pdf


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