Geodesic
The frequency and the structure of large character sums
Journal of the European Mathematical Society, Tome 20 (2018) no. 7, pp. 1759-1818 Cet article a éte moissonné depuis la source EMS Press

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Let M(χ) denote the maximum of ∣∑n≤N​χ(n)∣ for a given non-principal Dirichlet character χ modulo q, 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 modulo q, 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/2​χ(n)∣=π∣2−χ(2)∣​q​∣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 πq​​∣L(1,χ)∣.
DOI : 10.4171/jems/799
Classification : 11-XX
Keywords: Distribution of character sums, distribution of Dirichlet L-functions, pretentious multiplicative functions, random multiplicative functions 
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     author = {Jonathan Bober and Leo Goldmakher and Andrew Granville and Dimitris Koukoulopoulos},
     title = {The frequency and the structure of large character sums},
     journal = {Journal of the European Mathematical Society},
     pages = {1759--1818},
     year = {2018},
     volume = {20},
     number = {7},
     doi = {10.4171/jems/799},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/799/}
}
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Jonathan Bober; Leo Goldmakher; Andrew Granville; Dimitris Koukoulopoulos. The frequency and the structure of large character sums. Journal of the European Mathematical Society, Tome 20 (2018) no. 7, pp. 1759-1818. doi: 10.4171/jems/799

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