The Ramanujan Journal

, Volume 31, Issue 3, pp 271–279 | Cite as

A generalization of the Pólya–Vinogradov inequality

  • D. A. Frolenkov
  • K. Soundararajan


In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.


Arithmetic functions Dirichlet character Pólya–Vinogradov inequality 

Mathematics Subject Classification

11A25 11L40 



The First author is supported by the Dynasty Foundation, by the Russian Foundation for Basic Research (grants no. 11-01-00759-a and no. 12-01-31165). The second author is partially supported by NSF grant DMS-1001068.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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