On sums of Kloosterman and Gauss sums

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.