Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

Abstract

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd order character sums. Furthermore, conditionally on the Generalized Riemann Hypothesis we obtain a bound for odd order character sums which is best possible.

arXiv comment45 pages. Theorem 3 (and its proof in Section 9) added to this version

Cited by in corpus (2)

• On Character Sums and Exponential Sums over Generalized Arithmetic Progressions
• Character Sums Over The Prime Numbers