Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Goldmakher, Leo

doi:10.2140/ant.2012.6.123

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Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Leo Goldmakher

Vol. 6 (2012), No. 1, 123–163

DOI: 10.2140/ant.2012.6.123

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.

Keywords

Dirichlet characters, character sums, exponential sums, multiplicative functions

Mathematical Subject Classification 2000

Primary: 11L40

Secondary: 11L03, 11L07

Milestones

Received: 21 October 2010

Accepted: 29 December 2010

Published: 15 June 2012

Proposed: Andrew Granville

Seconded: Peter Sarnak

Authors

Leo Goldmakher
University of Toronto Department of Mathematics 40 St. George Street Toronto, ON M5S 2E4 Canada
http://www.math.toronto.edu/lgoldmak

Vol. 6, No. 1, 2012