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Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields

Stephan Baier, Peng Gao

TL;DR

The paper extends sharp upper-bound techniques for shifted moments of Dirichlet $L$-functions to a fixed modulus into the function-field setting. By adapting the Szabó framework and Soundararajan–Harper ideas, it combines truncated Dirichlet polynomials, explicit L-function bounds in function fields, and Perron-type formulas to bound $\sum_{\\chi\in X_Q^*} \prod_j |L(1/2+it_j,\\chi)|^{a_j}$ in terms of $\\varphi(Q)$, $(\\log|Q|)$, and a correlation factor given by $\\zeta_A$. A key outcome is Theorem $t1$, detailing sharp upper bounds for shifted moments and a corollary bounding moments of Dirichlet character sums $S_m(Q,Y)$ with $S_m(Q,Y) \ll \varphi(Q) Y^m (\\log|Q|)^{(m-1)^2}$. The results are unconditional in the function-field context (thanks to the function-field RH) and yield function-field analogues of known number-field results, with explicit dependence on the shifted parameters through the correlation factor $\\zeta_A(1+i(t_j-t_l)+1/\\log|Q|)$.

Abstract

In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet $L$-functions to a fixed modulus over function fields. We apply the result to obtain upper bounds on moments of Dirichlet character sums over function fields.

Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields

TL;DR

The paper extends sharp upper-bound techniques for shifted moments of Dirichlet -functions to a fixed modulus into the function-field setting. By adapting the Szabó framework and Soundararajan–Harper ideas, it combines truncated Dirichlet polynomials, explicit L-function bounds in function fields, and Perron-type formulas to bound in terms of , , and a correlation factor given by . A key outcome is Theorem , detailing sharp upper bounds for shifted moments and a corollary bounding moments of Dirichlet character sums with . The results are unconditional in the function-field context (thanks to the function-field RH) and yield function-field analogues of known number-field results, with explicit dependence on the shifted parameters through the correlation factor .

Abstract

In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet -functions to a fixed modulus over function fields. We apply the result to obtain upper bounds on moments of Dirichlet character sums over function fields.
Paper Structure (9 sections, 13 theorems, 61 equations)

This paper contains 9 sections, 13 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Backgrounds on function fields
  4. Sums over primes
  5. Perron’s formula
  6. Proof of Theorem \ref{['t1']}
  7. Various bounds for $L$-functions
  8. Completion of the proof
  9. Proof of Theorem \ref{['quadraticmean']}

Key Result

Theorem 1.1

Keep the notations above. Let $k\geq 1$ be a fixed integer and $a_1,\ldots, a_{k}$ be fixed positive real numbers. Then for any real $k$-tuple $t=(t_1,\ldots ,t_{k})$, we have where $L(s, \chi)$ is the $L$-function associated to $\chi$ defined in Ldef and where $\zeta_A(s)$ is the zeta function associated to $A$ defined in zetadef. Consequently, we have where we define $\overline {\theta}=\min_{

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.6
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 5 more