The frequency and the structure of large character sums

Abstract

Let M(\chi) denote the maximum of |\sum_{n\le N}\chi(n)| for a given non-principal Dirichlet character \chi modulo q , and let N_\chi denote a point at which the maximum is attained. In this article we study the distribution of M(\chi)/\sqrt{q} as one varies over characters modulo q , where q is prime, and investigate the location of N_\chi . We show that the distribution of M(\chi)/\sqrt{q} converges weakly to a universal distribution \Phi , uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \Phi 's tail. Almost all \chi for which M(\chi) is large are odd characters that are 1-pretentious. Now, M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)| , and one knows how often the latter expression is large, which has been how earlier lower bounds on \Phi were mostly proved. We show, though, that for most \chi with M(\chi) large, N_\chi is bounded away from q/2 , and the value of M(\chi) is little bit larger than \frac{\sqrt{q}}{\pi} |L(1,\chi)| .

Locations

• UCL Discovery (University College London) - View - PDF
• arXiv (Cornell University) - View - PDF
• Bristol Research (University of Bristol) - View - PDF
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• Journal of the European Mathematical Society - View