2024/05/01 by Alexander P. Mangerel, Mangerel, Alexander P., Y. You +1
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.00544
openalex publication_date 2024/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let χ be a primitive character modulo q, and let δ> 0. Assuming that χ has large order d, for any dth root of unity α we obtain non-trivial upper bounds for the number of n ≤ x such that χ(n) = α, provided x > qδ. This improves upon a previous result of the first author by removing restrictions on q and d. As a corollary, we deduce that if the largest prime factor of d satisfies P+(d) → ∞ then the level set χ(n) = α has o(x) such solutions whenever x > qδ, for any fixed δ> 0. Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters χ^ℓ, averaged over 1 ≤ ℓ ≤ d-1, due to the first author, which goes beyond Burgess' theorem as soon as d is sufficiently large. We in fact show the alternative result that either (a) the partial sum of χ itself, or (b) the partial sum of χ^ℓ, for ``almost all'' 1 ≤ ℓ ≤ d-1, exhibits cancellation on the interval [1,qδ], for any fixed δ> 0. By an analogous method, we also show that the Pólya-Vinogradov inequality may be improved for either χ itself or for almost all χ^ℓ, with 1 ≤ ℓ ≤ d-1. In particular, our averaged estimates are non-trivial whenever χ has sufficiently large even order d.