Large values of character sums with multiplicative coefficients

TL;DR

This paper investigates the magnitude of Dirichlet character sums with multiplicative coefficients, establishing lower bounds in specific ranges of N relative to the prime modulus q.

Contribution

It provides an Omega result demonstrating large values of these sums within a certain range of N, advancing understanding of their extremal behavior.

Findings

Proves an Omega lower bound for character sums with multiplicative coefficients.

Establishes the existence of large character sums in the specified range.

Contributes to the theory of character sums and their extremal properties.

Abstract

In this article, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove an Omega result in the region $\exp((\log q)^{\frac12+\varepsilon})\le N\le\sqrt q$, where $q$ is the prime modulus.