Geodesic
Large character sums: Burgess's theorem and zeros of $L$-functions
Journal of the European Mathematical Society, Tome 20 (2018) no. 1, pp. 1-14.

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We study the conjecture that ∑n≤x​χ(n)=o(x) for any primitive Dirichlet character χ modulo q with x≥qε, which is known to be true if the Riemann Hypothesis holds for L(s,χ). We show that it holds under the weaker assumption that „100%" of the zeros of L(s,χ) up to height 41​ lie on the critical line. We also establish various other consequences of having large character sums; for example, that if the conjecture holds for χ2 then it also holds for χ.
DOI : 10.4171/jems/757
Classification : 11-XX
Keywords: Bounds on character sums, zeros of Dirichlet L-functions, multiplicative functions 
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Andrew Granville; Kannan Soundararajan. Large character sums: Burgess's theorem and zeros of $L$-functions. Journal of the European Mathematical Society, Tome 20 (2018) no. 1, pp. 1-14. doi : 10.4171/jems/757. http://geodesic.mathdoc.fr/articles/10.4171/jems/757/

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