# Small G\'al sums and applications

## TL;DR

This paper explores small Gál sums in relation to the $L^1$-norm and applies the findings to improve bounds on character sums, non-vanishing theta functions, and moments of character sums.

## Contribution

It introduces the study of small Gál sums and their applications to refine bounds in analytic number theory problems.

## Key findings

- Refined logarithmic bounds on character sums.
- Improved lower bounds for non-vanishing theta functions.
- New lower bounds for moments of character sums.

## Abstract

In recent years, maximizing G\'al sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of G\'al type, with respect to the $L^1$-norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of non vanishing theta functions associated to Dirichlet characters; and (iii) new lower bounds for low moments of character sums.