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The second moment of cubic Dirichlet L-functions over function fields

Shivani Goel, Anwesh Ray

TL;DR

This work determines the second moment of cubic Dirichlet $L$-functions at the central point over the rational function field in the non-Kummer case $q \equiv 2 \pmod{3}$. Using Perron's formula and the approximate functional equation, the authors translate the moment into sums over polynomials with divisor weights, then carefully separate a dominant principal term (capturing cubes and non-cubes) from a dual term controlled by averages of Gauss sums. The main term yields an explicit asymptotic $\sim \frac{g(g+2) A_q(1/q^2,1/q^{3/2}) \zeta_q(3/2)^2}{8 \zeta_q(3)} q^{g+2}$, while the dual term is shown to be smaller and absorbed into the error. The results extend known first-moment formulas to a sharp second-moment asymptotic in function fields, relying on analytic techniques adapted to this setting and explicit Gauss-sum estimates.

Abstract

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result extends prior work of David, Florea and Lalin, who obtained an asymptotic formula for the first moment. Our approach relies on analytic techniques (Perron's formula, approximate functional equation, etc), adapted to the function field context. A key step in the construction is to relate second moment to certain averages of Gauss sums, which are estimated in loc. cit. using results of Kubota and Hoffstein.

The second moment of cubic Dirichlet L-functions over function fields

TL;DR

This work determines the second moment of cubic Dirichlet -functions at the central point over the rational function field in the non-Kummer case . Using Perron's formula and the approximate functional equation, the authors translate the moment into sums over polynomials with divisor weights, then carefully separate a dominant principal term (capturing cubes and non-cubes) from a dual term controlled by averages of Gauss sums. The main term yields an explicit asymptotic , while the dual term is shown to be smaller and absorbed into the error. The results extend known first-moment formulas to a sharp second-moment asymptotic in function fields, relying on analytic techniques adapted to this setting and explicit Gauss-sum estimates.

Abstract

In this article, we study the second moment of cubic Dirichlet L-functions at the central point over the rational function field , where is a power of an odd prime satisfying . Our result extends prior work of David, Florea and Lalin, who obtained an asymptotic formula for the first moment. Our approach relies on analytic techniques (Perron's formula, approximate functional equation, etc), adapted to the function field context. A key step in the construction is to relate second moment to certain averages of Gauss sums, which are estimated in loc. cit. using results of Kubota and Hoffstein.
Paper Structure (19 sections, 9 theorems, 152 equations)

This paper contains 19 sections, 9 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Methodology
  5. Organization
  6. Acknowledgement
  7. Preliminary notions
  8. Basic notation
  9. The zeta function
  10. Characters and L-polynomials
  11. The approximate functional equation
  12. Primitive characters of $\mathbb{F}_q[T]$
  13. Averages of Gauss sums
  14. Quadratic moments of cubic L-functions
  15. Setting up the problem
  16. ...and 4 more sections

Key Result

Theorem A

With respect to notation above, assume that $q \equiv 2 \pmod{3}$, then, where $A_q$ is the function defined in A_Qdefn.

Theorems & Definitions (19)

  • Theorem A
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3: Approximate functional equation
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 9 more