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 $\tau(\alpha) \gt (m+1)/2$.
Citation
Xiumin Ren. Yangbo Ye. "Hyper-Kloosterman Sums of Different Moduli and Their Applications to Automorphic Forms for $\operatorname{SL}_m(\mathbb{Z})$." Taiwanese J. Math. 20 (6) 1251 - 1274, 2016. https://doi.org/10.11650/tjm.20.2016.7389
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