The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound

Abstract

We prove a Lindelöf-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^{\ast }|d$, where $q^{\ast }$ is the least positive integer such that $q^2|(q^{\ast }{)}^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.

One sentence in section 1.5 deleted. To appear in Duke Math. J