On sums of Kloosterman and Gauss sums
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- by Igor E. Shparlinski PDF
- Trans. Amer. Math. Soc. 371 (2019), 8679-8697 Request permission
Abstract:
We present a new approach to bounding certain double sums of Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst these sums with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent estimates of Blomer, Fouvry, Kowalski, Michel, and Milićević and also of Fouvry, Kowalski, and Michel on double sums with Kloosterman sums. We also improve, in some ranges, a bound of Bettin and Chandee on certain triple sums with incomplete Kloosterman sums.
As the main application, we improve the error term, given by the above authors, in the asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms. We also give applications of our ideas to estimating cancellations amongst double Kloosterman sums.
References
Additional Information
- Igor E. Shparlinski
- Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
- MR Author ID: 192194
- Email: igor.shparlinski@unsw.edu.au
- Received by editor(s): August 11, 2017
- Received by editor(s) in revised form: January 9, 2018
- Published electronically: February 28, 2019
- Additional Notes: This work was supported by ARC Grant DP170100786.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8679-8697
- MSC (2010): Primary 11D79, 11L07
- DOI: https://doi.org/10.1090/tran/7506
- MathSciNet review: 3955560