Three conjectures about character sums

TL;DR

This paper demonstrates the equivalence of three major conjectures related to Dirichlet character sums and L-functions, and introduces a new mean value theorem for multiplicative functions, advancing understanding in analytic number theory.

Contribution

It proves the equivalence of three key conjectures about Dirichlet characters and L-functions, and establishes a new mean value theorem for multiplicative functions.

Findings

Three conjectures about Dirichlet characters are shown to be equivalent.

A new mean value theorem for weighted sums of multiplicative functions is derived.

Results connect different aspects of character sum bounds and L-function estimates.

Abstract

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estimate for short character sums, and upper bounds for $L(1,\chi)$ and $L(1+it,\chi)$) are more-or-less "equivalent". We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.