Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications

Abstract

A new reciprocity formula for Dirichlet $L$-functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$-functions in the $t$-aspect to the cubic moment of central $L$-values of Hecke-Maass newforms of level at most $q^2$ and primitive central character ${\psi }^2$ averaged over all primitive nonquadratic characters $\psi$ modulo $q$. Our formula can be thought of as a reverse version of recent work of Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval fourth moment bound and the twelfth moment bound for Dirichlet $L$-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.

arXiv comment43 pages. Open Access

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Cited by in corpus (3)

• Local integral transforms and global spectral decomposition
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions I: The Cuspidal Case
• Strong Hybrid Subconvexity for Twisted Selfdual ${\mathrm{GL}}_3$ $L$-Functions