2022/07/28 by Alexander P. Mangerel, Mangerel, Alexander P.
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2207.14377
openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let χ be a primitive character modulo a prime q, and let δ> 0. It has previously been observed that if χ has large order d ≥ d0(δ) then χ(n) ≠ 1 for some n ≤ qδ, in analogy with Vinogradov's conjecture on quadratic non-residues. We give a new and simple proof of this fact. We show, furthermore, that if d is squarefree then for any dth root of unity α the number of n ≤ x such that χ(n) = α is od → ∞(x) whenever x > qδ. Consequently, when χ has sufficiently large order the sequence (χ(n))n ≤ qδ cannot cluster near 1 for any δ> 0. Our proof relies on a second moment estimate for short sums of the characters χ^ℓ, averaged over 1 ≤ ℓ ≤ d-1, that is non-trivial whenever d has no small prime factors. In particular, given any δ> 0 we show that for all but o(d) powers 1 ≤ ℓ ≤ d-1, the partial sums of χ^ℓ exhibit cancellation in intervals n ≤ qδ as long as d ≥ d0(δ) is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime 3 ≤ d ≤ q-1, the Pólya-Vinogradov inequality may be improved for χ^ℓ on average over 1 ≤ ℓ ≤ d-1, extending work of Granville and Soundararajan.