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

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.