The fourth moment of Dirichlet $L$-functions at the central value

Xiaosheng Wu

Published 2020-08-31Version 1

The fourth moment of Dirichlet $L$-functions at the central point has been widely concerned, and an asymptotic formula with a power saving error term has been proved by Young for prime moduli, while the general moduli case is still open. In this work, we prove an asymptotic formula with a power saving error term for the fourth moment of Dirichlet $L$-functions at the central point for general moduli. The work relies on our deduction of the theory for a special divisor sum function, called $\mathcal{D}_q$-function, which is the key to deduce the main term and to apply the Kuznetsov trace formula for congruence groups. Another key ingredient is an uniform bound for a double sum in Kloosterman sums, which is of independent interest and may be useful elsewhere.