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Expected Values of $L$-functions Away from the Central Point

Chantal David, Patrick Meisner

TL;DR

This work computes the first moment of cubic Dirichlet $L$-functions over $\mathbb{F}_q[T]$ away from the central point, averaged over the thin family $\mathfrak{H}(3g)$. By developing a general approximate functional equation and splitting into principal and dual sums, the authors identify an explicit Euler-product main term $M_q(s)$ and, crucially, a pole-driven secondary term that undergoes a transition at $s=\tfrac{1}{3}$, with the dual sum canceling this secondary term for $s\neq\tfrac{1}{3}$. At $s=\tfrac{1}{3}$ the average grows linearly with the genus, $\sim C_q g$, and the random-matrix model predicts a weighted unitary statistic that agrees with these function-field results; in the limit $q\to\infty$, the first moment matches a unitary matrix integral weighted by $\det(1-\wedge^3 U)$. Overall, the paper provides a rigorous first moment computation at a non-central point, reveals a sharp transition in behavior, and connects the arithmetic of cubic characters to refined random-matrix predictions.

Abstract

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindelöf hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.

Expected Values of $L$-functions Away from the Central Point

TL;DR

This work computes the first moment of cubic Dirichlet -functions over away from the central point, averaged over the thin family . By developing a general approximate functional equation and splitting into principal and dual sums, the authors identify an explicit Euler-product main term and, crucially, a pole-driven secondary term that undergoes a transition at , with the dual sum canceling this secondary term for . At the average grows linearly with the genus, , and the random-matrix model predicts a weighted unitary statistic that agrees with these function-field results; in the limit , the first moment matches a unitary matrix integral weighted by . Overall, the paper provides a rigorous first moment computation at a non-central point, reveals a sharp transition in behavior, and connects the arithmetic of cubic characters to refined random-matrix predictions.

Abstract

We compute the expected value of Dirichlet -functions defined over attached to cubic characters evaluated at an arbitrary . We find a transition term at the point , reminiscent of the transition at the point of the bound for the size of an -function implied by the Lindelöf hypothesis. We show that at , the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.
Paper Structure (23 sections, 19 theorems, 207 equations)

This paper contains 23 sections, 19 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Setup and Main Result
  3. Random Matrix Model
  4. Weighted Random Matrix Model
  5. Structure of the paper
  6. Background on $L$-functions
  7. Cubic characters over $\mathbb{F}_q[T]$
  8. Functional Equation
  9. Principal and Dual Terms
  10. Perron's Formula
  11. The Principal Sum
  12. Contributions of the Principal Sum
  13. Error Term
  14. Main Term
  15. The Dual Term
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $q\equiv 1 \bmod{6}$ be a prime power, and $\epsilon>0$. Then for any $0<s<1, s \neq \frac{1}{3}$, we have where and $\zeta_q(s)$ is defined by def-zeta. If $s=\frac{1}{3}$, we have where the $O(1)$ is absolute (it does not depend on $q$ or $g$) as long at $g \geq 2$, $B_q$ is defined by def-Bq, and

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2: Approximate functional equation
  • proof
  • Proposition 2.3
  • Proposition 3.1
  • ...and 22 more