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.

arXiv commentOne sentence in section 1.5 deleted. To appear in Duke Math. J

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• Refinements to the prime number theorem for arithmetic progressions
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• On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case
• Subconvexity bounds for twisted L-functions, II
• Subconvexity for $L$-Functions on ${\mathrm{GL}}_3$ over Number Fields
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• Mixed Moments of the Riemann Zeta and Dirichlet $L$-Functions
• Strong Hybrid Subconvexity for Twisted Selfdual ${\mathrm{GL}}_3$ $L$-Functions
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• Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect
• Moments and non-vanishing of $L$-functions over thin subgroups
• Hybrid Weyl-type bound for $p$-power twisted $\mathrm{GL}(2)$ $L$-functions
• Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions
• Explicit Burgess Bound for Composite Moduli
• Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds
• A subconvex bound for twisted $L$-functions
• Burgess bounds for short character sums evaluated at forms II: the mixed case