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arXiv · 2608.05961

Bounds for moments of twisted quadratic characters of prime modulus

Peng Gao·Yuetong Zhao
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Abstract

We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $\chi_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the smoothed $m$-th moment for all integers $m\geq 4$. A matching lower bound for all even integers $m\geq 4$ shows that this bound is optimal.

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Peng Gao, Yuetong Zhao. 2026-08-06. Bounds for moments of twisted quadratic characters of prime modulus. https://arxiv.org/abs/2608.05961

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