[Paper Review] On the third moment of $L( frac{1}{2}, \chi_d)$ II: the number field case

Adrian Diaconu, Ian Whitehead|arXiv (Cornell University)|Apr 2, 2018

Analytic Number Theory Research9 references3 citations

TL;DR

This paper establishes a smoothed asymptotic formula for the third moment of quadratic Dirichlet L-functions at the central critical value in the number field setting. Using meromorphic continuation of a multiple Dirichlet series and functional equations tied to the D4 root system, the authors prove the existence of a secondary term of size $x^{3/4}$, with a precise residue computation at $s = 3/4$, and an error term of order $O(x^{2/3 + \delta})$ for any $\delta > 0$, confirming a long-standing conjecture on secondary terms in moments of L-functions.

ABSTRACT

We establish a smoothed asymptotic formula for the third moment of quadratic {D}irichlet $L$-functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size $x^{\frac{3}{4}}$. The error term in the asymptotic formula is on the order of $O(x^{\frac{2}{3}+\delta})$ for every $\delta > 0.$

Motivation & Objective

• To establish a smoothed asymptotic formula for the third moment of central values $L(\frac{1}{2}, \chi_d)$ over fundamental discriminants in number fields.
• To prove the existence of a secondary term of size $x^{3/4}$ in the asymptotic expansion, which had been conjectured but not rigorously established.
• To compute the exact residue at $s = 3/4$ in the meromorphic continuation of the generating series $Z_0(s)$, resolving a key component of the secondary term.
• To extend the framework of multiple Dirichlet series with functional equations isomorphic to the Weyl group of $D_4$ to analyze higher moments and their secondary terms.
• To remove prior assumptions on meromorphic continuation and polynomial growth, thereby establishing the result unconditionally under weaker hypotheses.

Proposed method

• The authors define a four-variable multiple Dirichlet series $Z(s_1,s_2,s_3,s_4; \chi_{a_2c^2}, \chi_{a_1c^1})$ with modified Euler factors at primes dividing $d$, which satisfies functional equations isomorphic to the Weyl group of the $D_4$ root system.
• They use Bochner's principle to establish meromorphic continuation of the series to $\mathbb{C}^4$, and deduce that $Z(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, s)$ has a simple pole at $s = \frac{3}{4}$.
• The series $Z_0(s)$ is obtained as a quadratic twist of this object, and its meromorphic continuation to $\Re(s) > \frac{2}{3}$ is established via spectral decomposition and convergence estimates.
• The residue at $s = \frac{3}{4}$ is computed via Eulerian factorization over odd primes, yielding a product over $p \neq 2$ of rational functions in $p^{-1/2}$, which simplifies to $P(p^{-1/2})$.
• The Mellin inversion formula is applied to the smoothed sum $\sum_d L(\frac{1}{2}, \chi_{2d})^3 W(d/x)$, shifting the line of integration to $\Re(s) = \frac{2}{3} + \delta$ to extract the main and secondary terms.
• Estimates on the Mellin transform $\hat{W}(s)$ and the growth of $Z_0(s)$ in vertical strips are used to control the error term, which is shown to be $O(x^{2/3 + \delta})$.

Experimental results

Research questions

• RQ1 Does a secondary term of size $x^{3/4}$ exist in the smoothed third moment of quadratic Dirichlet $L$-functions at the central point?
• RQ2 What is the exact residue of the generating series $Z_0(s)$ at $s = \frac{3}{4}$, and how does it relate to the arithmetic of odd primes?
• RQ3 Can the existence of such a secondary term be established unconditionally, without assuming meromorphic continuation or polynomial growth?
• RQ4 How do the functional equations of multiple Dirichlet series with $D_4$ symmetry explain the appearance of secondary terms in higher moments?
• RQ5 Is the secondary term at $x^{3/4}$ a generic feature of higher moments, and can it be predicted from group-theoretic structures?

Key findings

• The generating series $Z_0(s)$ has meromorphic continuation to $\Re(s) > \frac{2}{3}$, with a pole of order seven at $s = 1$ and a simple pole at $s = \frac{3}{4}$.
• The residue at $s = \frac{3}{4}$ is computed as $\frac{9}{256\pi^2} \cdot \frac{1}{4}(-181 + 128\sqrt{2}) \Gamma(\frac{1}{4})^4 \zeta(\frac{1}{2})^7 \cdot \prod_{p \neq 2} P(p^{-1/2})$, with $P(p^{-1/2})$ given explicitly.
• The smoothed third moment satisfies $\sum_{d^*} L(\frac{1}{2}, \chi_{2d})^3 W(d/x) = x Q_W(\log x) + \mathrm{Res}_{s=3/4} Z_0(s) \cdot \hat{W}(3/4) x^{3/4} + O_{\delta}(x^{2/3 + \delta})$ for any $\delta > 0$.
• The error term is $O_{\delta}(x^{2/3 + \delta})$, improving upon the prior best bound of $O(x^{3/4 + \varepsilon})$ from Young (2013).
• The secondary term is confirmed unconditionally, and its residue matches the prediction of Zhang (2016) under weaker assumptions, with full agreement on the product over odd primes.
• The result confirms that secondary terms of size $x^{3/4}$ arise from the $D_4$ functional equation structure and are not artifacts of the multiple Dirichlet series construction.