Skip to main content
SpringerLink
Log in

Character sums with smooth numbers

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We use the large sieve inequality for smooth numbers due to Drappeau et al. (Smooth-supported multiplicative functions in arithmetic progressions beyond the \(x^{1/2}\)-barrier, Preprint, 2017. arXiv:1704.04831), together with some other arguments, to improve their bounds on the frequency of pairs \((q,\chi )\) of moduli q and primitive characters \(\chi \) modulo q, for which the corresponding character sums with smooth numbers are large.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Drappeau, A. Granville, and X. Shao, Smooth-supported multiplicative functions in arithmetic progressions beyond the \(x^{1/2}\)-barrier, Mathematika 63 (2017), 895–918.

  2. A. Harper, Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers, Preprint, arXiv:1208.5992.

  3. A. Hildebrand, Integers free of large prime divisors in short intervals, Quart. J. Math. Oxford 36 (1985), 57–69.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Hildebrand and G. Tenenbaum, On integers free of large prime factors, Trans. Amer. Math. Soc. 296 (1986), 265–290.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. N. Huxley, Large values of Dirichlet polynomials. III, Acta Arith. 26 (1974), 435–444.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Providence, RI, 2004.

  7. M. Jutila, On Linnik’s constant, Math. Scand. 41 (1977), 45–62.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Dover Publ., NY, 2004.

    MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to Adam Harper for proving the argument used in Section 4 together with the generous permission to present it here as well as further comments. The author would also like to thank the anonymous referee for the careful reading of the manuscript and for many valuable suggestions. Some parts of this work were done when the author was visiting the Max Planck Institute for Mathematics, Bonn, and the Institut de Mathématiques de Jussieu, Université Paris Diderot, whose generous support and hospitality are gratefully acknowledged. This work was also partially supported by the Australian Research Council Grant DP170100786.

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Igor E. Shparlinski

Authors
  1. Igor E. Shparlinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor E. Shparlinski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shparlinski, I.E. Character sums with smooth numbers. Arch. Math. 110, 467–476 (2018). https://doi.org/10.1007/s00013-018-1168-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-018-1168-y

Mathematics Subject Classification

Keywords

Search

Navigation