TY - JOUR

T1 - The mean square of the product of the Riemann zeta-function with Dirichlet polynomials

AU - Bettin, Sandro

AU - Chandee, Vorrapan

AU - Radziwill, Maksym

N1 - Publisher Copyright:
© De Gruyter 2017.

PY - 2017/8

Y1 - 2017/8

N2 - 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 T1/2+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 ofWatt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to T3/4 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.

AB - 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 T1/2+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 ofWatt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to T3/4 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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U2 - 10.1515/crelle-2014-0133

DO - 10.1515/crelle-2014-0133

M3 - Article

AN - SCOPUS:85029211957

SN - 0075-4102

VL - 2017

SP - 51

EP - 79

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 729

ER -