Algebra & Number Theory

Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Leo Goldmakher

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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.

Article information

Source
Algebra Number Theory, Volume 6, Number 1 (2012), 123-163.

Dates
Received: 21 October 2010
Accepted: 29 December 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729759

Digital Object Identifier
doi:10.2140/ant.2012.6.123

Mathematical Reviews number (MathSciNet)
MR2950162

Zentralblatt MATH identifier
1263.11076

Subjects
Primary: 11L40: Estimates on character sums
Secondary: 11L03: Trigonometric and exponential sums, general 11L07: Estimates on exponential sums

Keywords
Dirichlet characters character sums exponential sums multiplicative functions

Citation

Goldmakher, Leo. Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality. Algebra Number Theory 6 (2012), no. 1, 123--163. doi:10.2140/ant.2012.6.123. https://projecteuclid.org/euclid.ant/1513729759


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References

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