4.5 Article

Large character sums: Burgess's theorem and zeros of L-functions

Journal

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
Volume 20, Issue 1, Pages 1-14

Publisher

EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/JEMS/757

Keywords

Bounds on character sums; zeros of Dirichlet L-functions; multiplicative functions

Funding

  1. NSERC
  2. Canadian Research Chair
  3. ERC
  4. NSF [DMS 1001068]
  5. Simons Foundation

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We study the conjecture that Sigma(n <= x) chi(n) = o(x) for any primitive Dirichlet character chi modulo q with x >= q(epsilon), which is known to be true if the Riemann hypothesis holds for L(s, chi). We show that it holds under the weaker assumption that 100% of the zeros of L(s, chi) up to height 1/4 lie on the critical line. We also establish various other consequences of having large character sums; for example, that if the conjecture holds for chi(2) then it also holds for chi.

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