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Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field

Hua Lin, Peng-Jie Wong

TL;DR

The paper advances Keating–Snaith’s conjecture for central L-values by establishing a conditional lower bound for a non-self-dual, non-rational family of cubic Hecke L-functions over the Eisenstein field under GRH. Building on the Radziwiłł–Soundararajan methodology, it combines a log-approximation via Dirichlet polynomials, moment calculations of these polynomials, and twisted 1-level density bounds obtained through explicit formulae and Poisson summation over $\mathbb Z[\omega]$. A key technical contribution is the twisted Gauss-sum analysis for conductors that are not prime powers, which enables precise control of low-lying zeros in the unitary family of cubic Hecke characters. The main result yields a quantitative Gaussian-type lower bound for the normalized log central values, representing the first such conditional bound in a non-rational, non-self-dual family and thus providing evidence toward Keating–Snaith in this new setting. The work reinforces the link between L-value distributions, zero-density phenomena, and random matrix theory predictions within a function-field–inspired, number-field framework.

Abstract

A famous conjecture of Keating and Snaith asserts that central values of $L$-functions in a given family admit a log-normal distribution with a prescribed mean and variance depending on the symmetry type of the family. Based on a recent work of Radziwill and Soundararajan, we obtain a conditional lower bound towards Keating-Snaith's conjecture for a "thin" family of cubic Hecke $L$-functions over the Eisenstein field. A key new input is certain twisted estimates of the 1-level density of zeros of cubic Hecke $L$-functions, extending the previous work of David and Güloğlu, under the Generalised Riemann Hypothesis.

Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field

TL;DR

The paper advances Keating–Snaith’s conjecture for central L-values by establishing a conditional lower bound for a non-self-dual, non-rational family of cubic Hecke L-functions over the Eisenstein field under GRH. Building on the Radziwiłł–Soundararajan methodology, it combines a log-approximation via Dirichlet polynomials, moment calculations of these polynomials, and twisted 1-level density bounds obtained through explicit formulae and Poisson summation over . A key technical contribution is the twisted Gauss-sum analysis for conductors that are not prime powers, which enables precise control of low-lying zeros in the unitary family of cubic Hecke characters. The main result yields a quantitative Gaussian-type lower bound for the normalized log central values, representing the first such conditional bound in a non-rational, non-self-dual family and thus providing evidence toward Keating–Snaith in this new setting. The work reinforces the link between L-value distributions, zero-density phenomena, and random matrix theory predictions within a function-field–inspired, number-field framework.

Abstract

A famous conjecture of Keating and Snaith asserts that central values of -functions in a given family admit a log-normal distribution with a prescribed mean and variance depending on the symmetry type of the family. Based on a recent work of Radziwill and Soundararajan, we obtain a conditional lower bound towards Keating-Snaith's conjecture for a "thin" family of cubic Hecke -functions over the Eisenstein field. A key new input is certain twisted estimates of the 1-level density of zeros of cubic Hecke -functions, extending the previous work of David and Güloğlu, under the Generalised Riemann Hypothesis.

Paper Structure

This paper contains 11 sections, 16 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Key propositions
  4. Proof of the main theorem
  5. Proofs of key propositions
  6. Proof of Proposition \ref{['Prop 1']}
  7. Proof of Proposition \ref{['Prop 2']}
  8. Proof of Proposition \ref{['Prop 3']}
  9. Twisted zero-sum estimates
  10. $S_2$ sum estimate
  11. Completing the proof of Proposition \ref{['Prop 3']}

Key Result

Theorem 1.1

Assume the Generalised Riemann Hypothesis (GRH) for $L(s, \chi_\mathfrak{f})$ with $\mathfrak{f}\in\mathcal{F}$. For any fixed $(\alpha,\beta)$, we have as $X \rightarrow \infty$, where the integrand is the probability density function of a standard normal random variable.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark
  • Lemma 2.1
  • Lemma 2.2: Explicit Formula
  • Lemma 2.3: Poisson Summation Formula
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.5
  • proof
  • ...and 17 more