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Sums of certain fractional parts

Abstract

In this note, an upper bound for the sum of fractional parts of certain smooth functions is given. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl’s bound for exponential sums and a device used by Popov.

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Acknowledgements

The author gratefully acknowledges the anonymous referee for some corrections and remarks that have significantly improved the paper.

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Correspondence to Olivier Bordellès.

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Bordellès, O. Sums of certain fractional parts. Period Math Hung 84, 203–210 (2022). https://doi.org/10.1007/s10998-021-00399-6

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  • DOI: https://doi.org/10.1007/s10998-021-00399-6

Keywords

  • Weyl’s and van der Corput’s exponential sums
  • Fractional part

Mathematics Subject Classification

  • Primary 11L07
  • Secondary 11L15
  • 11J54