On the cubic Weyl sum

2021/01/20 by Bryce Kerr, Kerr, Bryce
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.08058

openalex publication_date 2021/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any ε>0 there exists some δ>0 such that for any coprime integers a,q and real number γ we have ∑_{1≤ n ≤ N}e((an³)/(q)+γn)≪ (qN)^1/4 q^-δ, provided q^1/3+ε≤ N ≤ q^1/2-ε. Our argument builds on some ideas of Enflo.

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