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Motohashi’s fourth moment identity for non-archimedean test functions and applications

Part of: Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  17 April 2020

Valentin Blomer ,
Peter Humphries ,
Rizwanur Khan  and
Micah B. Milinovich
Valentin Blomer
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, D-53115Bonn, Germany email blomer@math.uni-bonn.de
Peter Humphries
Affiliation:
Department of Mathematics, University College London, Gower Street, LondonWC1E 6BT, UK email pclhumphries@gmail.com
Rizwanur Khan
Affiliation:
Department of Mathematics, University of Mississippi, University, MS38677, USA email rrkhan@olemiss.edu
Micah B. Milinovich
Affiliation:
Department of Mathematics, University of Mississippi, University, MS38677, USA email mbmilino@olemiss.edu
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Abstract

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic $L$-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet $L$-functions modulo $q$ weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length $q^{1/4}$. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic $L$-functions, which we also use to improve the best known subconvexity bounds for automorphic $L$-functions in the level aspect.

Keywords

spectral reciprocitymoments of L-functionssubconvexity

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 5 , May 2020 , pp. 1004 - 1038
Copyright
© The Authors 2020

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Footnotes

The first author is supported in part by DFG grant BL 915/2-2. The second author is supported by the European Research Council grant agreement 670239. The third author is supported by the Simons Foundation (award 630985).

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