Abstract
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all \({N \geq 2}\), satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
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Kowalski, E., Ricotta, G. Fourier coefficients of GL(N) automorphic forms in arithmetic progressions. Geom. Funct. Anal. 24, 1229–1297 (2014). https://doi.org/10.1007/s00039-014-0296-1
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DOI: https://doi.org/10.1007/s00039-014-0296-1
Keywords and phrases
- Voronoĭ summation formula
- generalized Bessel transforms
- fourier coefficients of GL(N) Hecke–Maass cusp forms
- arithmetic progressions
- central limit theorem
- hyper-kloosterman sums
- monodromy group
- sato–Tate equidistribution