Sharp bounds for maximal sums of odd order Dirichlet characters

2025/05/12 by Alexander P. Mangerel, Mangerel, Alexander P.
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.07651

openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g ≥ 3 be fixed and odd, and for large q let χ be a primitive Dirichlet character modulo q of order g. Conditionally on GRH we improve the existing upper bounds in the Pólya-Vinogradov inequality for χ, showing that M(χ) := max_{t ≥ 1} |∑_{n ≤ t} χ(n) | ≪ √(q) \frac(loglog q)^1-δ_g (logloglogloglog q)^δ_g(logloglog q)^1/4, where δ_g := 1-\tfracgπsin(π/g). Furthermore, we show unconditionally that there is an infinite sequence of order g primitive characters χ_j modulo q_j for which M(χ_j) ≫ √(q_j) \frac(loglog q_j)^1-δ_g (logloglogloglog q_j)^δ_g(logloglog q_j)^1/4, so that our GRH bound is sharp up to the implicit constant. This improves on previous work of Granville and Soundararajan, of Goldmakher, and of Lamzouri and the author.

Citations

• Long large character sums
• Large odd order character sums and improvements of the\n P 'olya-Vinogradov inequality
• Large even order character sums
• Multiplicative mimicry and improvements of the Polya-Vinogradov\n inequality

Related

• Large odd order character sums and improvements of the\n P 'olya-Vinogradov inequality
• Partial Gaussian sums and the Pólya--Vinogradov inequality for primitive characters
• Large sums of high order characters II
• Ω-bounds for the partial sums of some modified Dirichlet characters II
• Short Character Sums and the Pólya-Vinogradov Inequality