A lower bound on high moments of character sums

2024/09/20 by Barnabás Szabó, Szabó, Barnabás · 3 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.13436

openalex publication_date 2024/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any real k≥ 2 and large prime q, we prove a lower bound on the 2k-th moment of the Dirichlet character sum (1)/(ϕ(q)) ∑_\substackχ mod q
χ≠ χ₀ | ∑_{n≤ x} χ(n)|^2k, where 1≤ x≤ q, and χ is summed over the set of non-trivial Dirichlet characters mod q. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method.

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