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Upper bounds for shifted moments of quadratic Dirichlet $L$-functions over function fields

Peng Gao, Liangyi Zhao

TL;DR

This work establishes sharp unconditional upper bounds for shifted moments of quadratic Dirichlet L-functions over function fields, mirroring recent number-field results within the function-field setting where RH is known. The authors bound |L(1/2+it_j, χ_D)| via short Dirichlet polynomials in primes, carefully accounting for prime-square contributions, and deploy a Harper–Soundararajan–type framework to control high-moment growth. The main result yields explicit upper bounds that feature products of zeta_A factors reflecting off-diagonal interactions, and it is then used to bound moments of quadratic character sums S_m(q, g, Y). Additionally, Perron-type formulae in function fields facilitate translating sums into contour integrals, enabling uniform treatment across shifts t_j. Overall, the paper advances understanding of shifted moments in the function-field setting and provides tools for bounding related character-sum moments with potential connections to density conjectures.

Abstract

We establish sharp upper bounds on shifted moments of quadratic Dirichlet $L$-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.

Upper bounds for shifted moments of quadratic Dirichlet $L$-functions over function fields

TL;DR

This work establishes sharp unconditional upper bounds for shifted moments of quadratic Dirichlet L-functions over function fields, mirroring recent number-field results within the function-field setting where RH is known. The authors bound |L(1/2+it_j, χ_D)| via short Dirichlet polynomials in primes, carefully accounting for prime-square contributions, and deploy a Harper–Soundararajan–type framework to control high-moment growth. The main result yields explicit upper bounds that feature products of zeta_A factors reflecting off-diagonal interactions, and it is then used to bound moments of quadratic character sums S_m(q, g, Y). Additionally, Perron-type formulae in function fields facilitate translating sums into contour integrals, enabling uniform treatment across shifts t_j. Overall, the paper advances understanding of shifted moments in the function-field setting and provides tools for bounding related character-sum moments with potential connections to density conjectures.

Abstract

We establish sharp upper bounds on shifted moments of quadratic Dirichlet -functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.
Paper Structure (9 sections, 10 theorems, 84 equations)

This paper contains 9 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Backgrounds on function fields
  4. Estimates of various sums
  5. Perron’s formula
  6. Proof of Theorem \ref{['t1']}
  7. Initial treatment
  8. Completion of the proof
  9. Proof of Theorem \ref{['quadraticmean']}

Key Result

Theorem 1.1

With the notation as above, suppose that $X=q^{2g+1}$. Let $k\geq 1$ be a fixed integer, $a_1,\ldots, a_{k}$ fixed positive real numbers, and $t=(t_1,\ldots ,t_{k})$ be a real $k$-tuple. Then we have for $g$ larger than some constant depending on the $a_j$, Consequently, where $\overline {\theta}=\min_{n \in \mathbb Z} |\theta-2\pi n|$ for any $\theta \in \mathbb{R}$. The implied constants above

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 2 more