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

Ian Petrow, Matthew P. Young|arXiv (Cornell University)|Aug 27, 2019

Analytic Number Theory Research5 citations

TL;DR

This paper establishes a Weyl-strength subconvex bound for all Dirichlet $L$-functions at the central point, removing the prior cube-free conductor restriction. By proving a Lindelöf-on-average bound for the fourth moment of $L$-functions along a coset of Dirichlet characters, the authors extend their prior work using advanced automorphic forms and duality in the $q$-aspect, ultimately achieving the optimal $O(q^{1/6+ uarepsilon})$ subconvexity 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.

Motivation & Objective

• To establish a Weyl-strength subconvex bound for all primitive Dirichlet $L$-functions, regardless of conductor structure.
• To remove the cube-free conductor hypothesis that previously limited the applicability of the authors' prior results on cubic moments.
• To prove a Lindelöf-on-average upper bound for the fourth moment of $L$-functions along a coset of characters modulo $d$, where $q^* mid d$.
• To extend the duality method in the $q$-aspect to handle non-cube-free conductors by analyzing singular character contributions.
• To complete the subconvexity program for Dirichlet $L$-functions by unifying the treatment of archimedean and non-archimedean contributions in the moment method.

Proposed method

• Applies the Bruggeman-Kuznetsov formula and Poisson summation to transform the cubic moment into a dual moment involving a character sum $g( heta, ho)$.
• Uses the duality relation $\sum_{\psi \pmod{q}} |L(1/2, \psi)|^4 g(\chi, \psi) \ll q^{2+\varepsilon}$ as the central analytic estimate.
• Analyzes the character sum $g(\chi, \psi)$ via multiplicative structure and $p$-adic methods, especially for $q = p^3$, where singular characters arise.
• Applies $\ell$-adic sheaf theory and Deligne’s Riemann Hypothesis to bound $g(\chi, \psi)$ for $q = p$, and uses elementary estimates for $q = p^2$.
• Handles the Eisenstein series contribution by isolating polar terms from twisted $L$-functions and bounding them via Mellin inversion and spectral decomposition.
• Employs a dyadic partition of unity and spectral decomposition to control the sum over $h \equiv 0 \pmod{d}$, using bounds on $S_{\text{Maass}}, S_{\text{hol}}, S_{\text{Eis}}$, and $S_{\text{Pole}}$.

Experimental results

Research questions

• RQ1 Can the Weyl bound $L(1/2 + it, \chi) \ll (q(1+|t|))^{1/6+\varepsilon}$ be established for all primitive Dirichlet characters $\chi$ modulo $q$, without assuming $q$ is cube-free?
• RQ2 What is the behavior of the fourth moment $\sum_{\psi \pmod{q}} |L(1/2, \psi)|^4 g(\chi, \psi)$ when $q$ is not cube-free, particularly for $q = p^3$?
• RQ3 How do the singular characters $\psi$ modulo $q = p^3$ (with $p \equiv 1 \pmod{4}$) affect the duality and moment estimates?
• RQ4 Can the positivity argument based on $L(1/2, \pi \otimes \chi) \geq 0$ be extended beyond the cube-free case using the new moment bounds?
• RQ5 What is the precise size of the polar contribution $S_{\text{Pole}}$ in the Eisenstein series term, and does it remain within the required error tolerance?

Key findings

• The paper establishes the Weyl bound $L(1/2 + it, \chi) \ll_{\varepsilon} (q(1+|t|))^{1/6+\varepsilon}$ for all primitive Dirichlet characters $\chi$ modulo $q$, without any restriction on the conductor.
• The authors prove a Lindelöf-on-average bound for the fourth moment: $\sum_{\psi \pmod{q}} |L(1/2, \psi)|^4 g(\chi, \psi) \ll_{\varepsilon} q^{2+\varepsilon}$, which holds even when $q$ is not cube-free.
• For $q = p^3$ with $p \equiv 1 \pmod{4}$, there are $2(p-1)$ singular characters $\psi$ for which $|g(\chi, \psi)| = p^{1/2} q$, and these are handled via a refined decomposition of the sum.
• The polar contribution $S_{\text{Pole}}(\chi)$ is bounded by $O(d^{-1/8} \cdot \frac{NH}{q} (Nq)^\varepsilon)$, which is sufficiently small to not affect the main term.
• The bounds for the Maass, holomorphic, and Eisenstein contributions are unified via a dyadic decomposition and spectral analysis, with all terms controlled by $O((Nq)^\varepsilon N q^{-1} H)$.
• The final estimate confirms that the sum over $h \equiv 0 \pmod{d}$ in the spectral sum is bounded by $O((Nq)^\varepsilon N q^{-1} H)$, completing the proof of Theorem 1.5 and thus Theorem 1.1.