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

Abstract

We study the conjecture that \sum_{n\leq x} \chi(n)=o(x) for any primitive Dirichlet character \chi modulo q with x\geq 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 \tfrac 14 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 .