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

We study the conjecture that ${\sum }_{n\le x}\chi (n)=o(x)$ for any primitive Dirichlet character $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$ with $x\ge q^{ϵ}$, 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 `$100\mathrm{\%}$’ of the zeros of $L(s,\chi )$ up to height $\frac{1}{4}$ lie on the critical line; and establish various other consequences of having large character sums.