The frequency and the structure of large character sums

Jonathan Bober, Leo Goldmakher, Andrew Granville, Dimitris Koukoulopoulos

Research output: Contribution to journalArticle (Academic Journal)peer-review

8 Citations (Scopus)
252 Downloads (Pure)


Let M(χ) denote the maximum of |∑n≤Nχ(n)| for a given non-principal Dirichlet character χ(modq), and let Nχ denote a point at which the maximum is attained. In this article we study the distribution of M(χ)/q√ as one varies over characters (modq), where q is prime, and investigate the location of Nχ. We show that the distribution of M(χ)/q√ converges weakly to a universal distribution Φ, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for Φ's tail. Almost all χ for which M(χ) is large are odd characters that are 1-pretentious. Now, M(χ)≥|∑n≤q/2χ(n)|=|2−χ(2)|πq√|L(1,χ)|, and one knows how often the latter expression is large, which has been how earlier lower bounds on Φ were mostly proved. We show, though, that for most χ with M(χ) large, Nχ is bounded away from q/2, and the value of M(χ) is little bit larger than q√π|L(1,χ)|.
Original languageEnglish
Pages (from-to)1759-1818
Number of pages60
JournalJournal of the European Mathematical Society
Issue number7
Early online date22 May 2018
Publication statusPublished - 22 May 2018


  • math.NT
  • Primary: 11N60, Secondary: 11K41, 11L40


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