Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

Details Multiplicative mimicry and improvements of the Polya-Vinogradov inequality Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

2009-11-30

arxiv.org/abs/0911.5547v2

Mathematics

Number Theory

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

Scientific paper

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.

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