Short Character Sums and the Pólya-Vinogradov Inequality

2019/05/22 by Mangerel, Alexander P. · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1905.09238

Abstract

We show in a quantitative way that any odd character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya-Vinogradov inequality for χ can be improved to max_{1 ≤ t ≤ q} |∑_{n ≤ t} χ(n)| = o_{q → ∞}(√(q)log q) then for any ε> 0 one may exhibit cancellation in partial sums of χ on the interval [1,t] whenever t > q^ε, i.e.,∑_{n ≤ t} χ(n) = o_{q → ∞}(t) for all t gt; q^ε. This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya-Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.

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