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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On sums of Kloosterman and Gauss sums

Author: Igor E. Shparlinski
Journal: Trans. Amer. Math. Soc. 371 (2019), 8679-8697
MSC (2010): Primary 11D79, 11L07
Published electronically: February 28, 2019
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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.

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Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia

Keywords: Kloosterman sum, cancellation, double sum, trilinear sum
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.
Article copyright: © Copyright 2019 American Mathematical Society