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

  • Valentin Blomer (a1), Peter Humphries (a2), Rizwanur Khan (a3) and Micah B. Milinovich (a4)


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.



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

  • Valentin Blomer (a1), Peter Humphries (a2), Rizwanur Khan (a3) and Micah B. Milinovich (a4)


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