The mollified fourth moment of Dirichlet $L$-functions

TL;DR

This work establishes a power-saving asymptotic for the mollified fourth moment of Dirichlet $L$-functions across general moduli $q$ with $q\not\equiv 2\pmod 4$, using a mollifier of length up to $q^{1/22-\varepsilon}$. The authors develop a comprehensive framework combining approximate functional equations, orthogonality, Voronoi-type summation via the generalized Estermann $D$-function, and a delicate diagonal/off-diagonal analysis that leverages new bounds for sums of Kloosterman sums and bilinear forms of incomplete Kloosterman sums. A key innovation is an enhanced treatment of balanced and unbalanced terms, with the balanced portion handled by an advanced spectral large sieve and Kuznetsov-style analysis, and the unbalanced portion controlled by bilinear sum bounds (KSWX23). As a corollary, the mollified moment results sharpen Wu’s previous asymptotics, and the non-mollified fourth moment at the central point receives a significant power-saving improvement. The methods have potential applications to non-vanishing, large-value, and higher-morder moment problems for Dirichlet $L$-functions on general moduli.

Abstract

We prove an asymptotic formula with a power saving error term for the fourth moment of the family of Dirichlet $L$-functions to modulus $q$ mollified by a Dirichlet polynomial of length $q^{\frac1{22}-\ve}$, valid for all moduli $q\not\equiv2 \pmod 4$. This result was previously known only for restricted sets of moduli with smaller power savings. As a special case, when no Dirichlet polynomial is enclosed, this leads to a significant improvement on X. Wu's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions at the cental point.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Remark
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Approximate functional equation
• Lemma 2.2: The orthogonality formula
• Lemma 2.3: Smooth dyadic partition of unity
• Lemma 2.4
• Lemma 2.5