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 $q^{\ast }\mid d$, where $q^{\ast }$ is the least positive integer such that $q^2\mid {(q^{\ast })}^3$. 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.