Explicit Burgess inequalities for cubefree moduli

2025/11/21 by Hasanalizade, Elchin, Lin, Hua, Martin, Greg +2
Computer Science · Mathematics · #11L40 Primary #11Y60 Secondary #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.17778

openalex publication_date 2025/11/21 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28

Abstract

Burgess proved that for χ_q a primitive Dirichlet character modulo q with q cubefree, |∑_{M< n≤ M+N}χ_q(n)| ≪ N^1-(1)/(r)q^{(r+1)/(4r2)+ε} for all integers r≥1. 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≥ 3.

Citations

• Explicit Burgess Bound for Composite Moduli
• On Grosswald's conjecture on primitive roots
• The Burgess inequality and the least k-th power non-residue

Related

• An Investigation Into Several Explicit Versions of Burgess' Bound
• Large character sums: Burgess's theorem and zeros of L-functions
• Explicit Burgess Bound for Composite Moduli
• Explicit bounds for small prime nonresidues
• A refinement of the Burgess bound for character sums