Archiv der Mathematik

, Volume 110, Issue 5, pp 467–476 | Cite as

Character sums with smooth numbers

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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.

Keywords

Character sum Smooth number Large sieve 

Mathematics Subject Classification

Primary 11L40 Secondary 11N25 11N36 

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Notes

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.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of New South WalesSydneyAustralia

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