Vol. 6, No. 1, 2012

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Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Leo Goldmakher

Vol. 6 (2012), No. 1, 123–163

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.

Dirichlet characters, character sums, exponential sums, multiplicative functions
Mathematical Subject Classification 2000
Primary: 11L40
Secondary: 11L03, 11L07
Received: 21 October 2010
Accepted: 29 December 2010
Published: 15 June 2012

Proposed: Andrew Granville
Seconded: Peter Sarnak
Leo Goldmakher
University of Toronto
Department of Mathematics
40 St. George Street
Toronto, ON  M5S 2E4