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 * |d, where q * is the least positive integer such that q 2 |(q * ) 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.

The Weyl bound for Dirichlet $L$-functions of cube-free conductor

“…Remark added August 28, 2019: In [PY2], written after the first version of the present paper, the authors have extended all the cubic moment bounds stated in Section 1.1 to hold for arbitrary q. More precisely, [PY2] contains proofs of Conjectures 6.6 and 8.2 from the present paper, which are shown here to imply the cubic moment bounds for general q.…”

On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

“…Since the cubic moment for L-functions of Eisenstein series is the sixth moment of some Dirichlet L-functions, their result implies the Weyl-type subconvex bounds in the level aspect for Dirichlet L-functions of quadratic characters. Their method was recently refined and generalized by Petrow-Young to obtain the hybrid Weyl for cube-free level Dirichlet characters [24], then for all level Dirichlet characters [23]. Note that such methods are based on approximating the cubic moment side via the approximate functional equation and the Petersson-Kuznetsove formula, hence no explicit formula in the inverse direction of Motohashi's formula has been obtained.…”

Refinements to the prime number theorem for arithmetic progressions

“…. Without assuming bounds for Dirichlet L-functions beyond the recently proven hybrid Weyl bound [17], this appears to be the limit of our method when (β 1 , χ 1 ) exists. See Remark 5.8.…”

“…The desired result follows. It remains to choose φ = 1 6 + 10 −7 in (5.8), which is permissible by recent work of Petrow and Young [17]. With this choice, (5.18) holds trivially for 1/2 ≤ σ ≤ 3/5 because of the trivial bound N * q (σ, T ) ≪ ϕ(q)T log(qT ).…”