The eighth moment of Dirichlet $L$-functions

We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2) L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24024 appears as a factor in the leading order term of the eighth moment.

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