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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
-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet
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
. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic
-functions, which we also use to improve the best known subconvexity bounds for automorphic
-functions in the level aspect.
We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan
-Dirichlet series. Invent. Math.94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form
-functions of arbitrary conductor.
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