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Explicit Burgess inequalities for cubefree moduli

Elchin Hasanalizade, Hua Lin, Greg Martin, Andradis Luna Martínez, Enrique Treviño

TL;DR

This paper proves explicit Burgess-type bounds for Dirichlet character sums modulo cubefree moduli, extending beyond prime moduli to composite ones. The authors establish an explicit Weil-type inequality for short, shifted character sums and embed it in a general inductive framework to bound $|S_{\chi}(M,N)|$; they carefully choose and bound auxiliary parameters through a detailed analysis of $v_{\mathcal{A}}(x)$ sums and related arithmetic sums. The main contributions are an improved explicit constant for $r=2$ and explicit Burgess bounds for all $r\ge3$ when $q$ is cubefree, with the constants $C(r)$ and $D(r)$ computed (and tabulated) and two bound families $T_1(q)$ and $T_2(q)$ defined. These results provide practically usable explicit estimates for composite moduli, with potential applications to least $k$-th power residues, $L$-value bounds, and related analytic-number-theory problems.

Abstract

Burgess proved that for $χ_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\Big|\sum_{M< n\le M+N}χ_q(n)\Big| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+ε}$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$.

Explicit Burgess inequalities for cubefree moduli

TL;DR

This paper proves explicit Burgess-type bounds for Dirichlet character sums modulo cubefree moduli, extending beyond prime moduli to composite ones. The authors establish an explicit Weil-type inequality for short, shifted character sums and embed it in a general inductive framework to bound ; they carefully choose and bound auxiliary parameters through a detailed analysis of sums and related arithmetic sums. The main contributions are an improved explicit constant for and explicit Burgess bounds for all when is cubefree, with the constants and computed (and tabulated) and two bound families and defined. These results provide practically usable explicit estimates for composite moduli, with potential applications to least -th power residues, -value bounds, and related analytic-number-theory problems.

Abstract

Burgess proved that for a primitive Dirichlet character modulo with cubefree, for all integers More recently, explicit versions with prime moduli were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least -th power residue, and bounding the size of Dirichlet -functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for We improve their explicit constant for and compute an explicit Burgess bound for cubefree for .

Paper Structure

This paper contains 6 sections, 18 theorems, 128 equations, 1 table.

Table of Contents

  1. Introduction
  2. Explicit Weil-type inequality
  3. General setup
  4. The $v_{{\mathcal{A}}}$ sums
  5. Important bounds
  6. Proof of Main Theorems

Key Result

Theorem 1.1

Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo $q$. Let $C(r)$ be defined as in Table Table 1. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table Table 1 such that

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Weil-type inequality
  • Lemma 2.1: Burgess
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['weil']}
  • ...and 27 more