## Taiwanese Journal of Mathematics

### Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$

#### Abstract

Hyper-Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $\operatorname{GL}_m(\mathbb{Z})$. In this paper their square moment is evaluated and their bounds are proved in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of a Maass form for $\operatorname{SL}_m(\mathbb{Z})$ against an exponential function $e(\alpha n)$ are estimated. These sums are proved to have rapid decay when $\alpha$ is a fixed rational number or a transcendental number with approximation exponent $\tau(\alpha) \gt m$. Non-trivial bounds are proved for these sums when $\tau(\alpha) \gt (m+1)/2$.

#### Article information

Source
Taiwanese J. Math., Volume 20, Number 6 (2016), 1251-1274.

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874530

Digital Object Identifier
doi:10.11650/tjm.20.2016.7389

Mathematical Reviews number (MathSciNet)
MR3580294

Zentralblatt MATH identifier
1357.11068

#### Citation

Ren, Xiumin; Ye, Yangbo. Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$. Taiwanese J. Math. 20 (2016), no. 6, 1251--1274. doi:10.11650/tjm.20.2016.7389. https://projecteuclid.org/euclid.twjm/1498874530

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