Advertisement

The Ramanujan Journal

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

A generalization of the Pólya–Vinogradov inequality

  • D. A. Frolenkov
  • K. Soundararajan
Article

Abstract

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.

Keywords

Arithmetic functions Dirichlet character Pólya–Vinogradov inequality 

Mathematics Subject Classification

11A25 11L40 

Notes

Acknowledgements

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.

References

  1. 1.
    Bachman, G., Rachakonda, L.: On a problem of Dobrowolski and Williams and the Pólya–Vinogradov inequality. Ramanujan J. 5, 65–71 (2001) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Burgess, D.A.: On a conjecture of Norton. Acta Arith. 27, 265–267 (1975) MathSciNetMATHGoogle Scholar
  3. 3.
    Dobrowolski, E., Williams, K.S.: An upper bound for the sum \(\sum_{n=a+1}^{a+H}f(n)\) for a certain class of functions f. Proc. Am. Math. Soc. 114, 29–35 (1992) MathSciNetMATHGoogle Scholar
  4. 4.
    Frolenkov, D.A.: A numerically explicit version of the Pólya–Vinogradov inequality. Mosc. J. Comb. Number Theory 1(3), 25–41 (2011) MathSciNetMATHGoogle Scholar
  5. 5.
    Granville, A., Soundararajan, K.: Large character sums: pretentious characters and the Pólya–Vinogradov theorem. J. Am. Math. Soc. 20(2), 357–384 (2007) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hildebrand, A.: On the constant in the Pólya–Vinogradov inequality. Can. Math. Bull. 31, 347–352 (1988) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Montgomery, H.L.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS, vol. 84. AMS, Providence (1994) MATHGoogle Scholar
  8. 8.
    Pomerance, C.: Remarks on the Pólya–Vinogradov Inequality Integers (Proceedings of the Integers Conference, October 2009), 11A (2011), Article 19, 11p Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

Personalised recommendations