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

Abstract

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.

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