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

TL;DR

This paper investigates the behavior of large character sums for primitive Dirichlet characters, showing that certain zero distribution assumptions of $L$-functions imply the conjectured bounds, with implications for number theory.

Contribution

It demonstrates that the conjecture on character sums holds under the assumption that all zeros up to a certain height lie on the critical line, a weaker condition than the Riemann Hypothesis.

Findings

The conjecture holds if 100% of zeros up to height 1/4 are on the critical line.

Establishes consequences of large character sums under zero distribution assumptions.

Links zero distribution of $L$-functions to bounds on character sums.

Abstract

We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ 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; and establish various other consequences of having large character sums.