The frequency and the structure of large character sums

  • Jonathan Bober

    University of Bristol, UK
  • Leo Goldmakher

    Williams College, Williamstown, USA
  • Andrew Granville

    Université de Montréal, Canada
  • Dimitris Koukoulopoulos

    Université de Montréal, Canada

Abstract

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

Cite this article

Jonathan Bober, Leo Goldmakher, Andrew Granville, Dimitris Koukoulopoulos, The frequency and the structure of large character sums. J. Eur. Math. Soc. 20 (2018), no. 7, pp. 1759–1818

DOI 10.4171/JEMS/799