### Abstract

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 language | English |
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Pages (from-to) | 1759-1818 |

Number of pages | 60 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Issue number | 7 |

Early online date | 22 May 2018 |

DOIs | |

Publication status | Published - 22 May 2018 |

### Keywords

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

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## Profiles

## Dr Jonathan W Bober

- School of Mathematics - Heilbronn Senior Research Fellow
- Heilbronn Institute for Mathematical Research
- Pure Mathematics

Person: Academic , Member

## Cite this

Bober, J., Goldmakher, L., Granville, A., & Koukoulopoulos, D. (2018). The frequency and the structure of large character sums.

*Journal of the European Mathematical Society*,*20*(7), 1759-1818. https://doi.org/10.4171/JEMS/799