# Kloosterman paths of prime powers moduli

### Guillaume Ricotta

Université de Bordeaux I, Talence, France### Emmanuel Royer

Université Clermont Auvergne, Aubière, France

## Abstract

In [12], the authors proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b_{0};p)/p_{1/2}$ converge in the sense of finite distributions to a specific random Fourier series, as $a$ varies over $(Z/pZ)_{×}$, $b_{0}$ is fixed in $(Z/pZ)_{×}$ and $p$ tends to infinity among the odd prime numbers. This article considers the case of $S(a,b_{0};p_{n})/p_{n/2}$, as $a$ varies over $(Z/p_{n}Z)_{×}$, $b_{0}$ is fixed in $(Z/p_{n}Z)_{×}$, $p$ tends to infinity among the odd prime numbers and $n≥2$ is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on $[0,1]$ is also established, as $(a,b)$ varies over $(Z/p_{n}Z)_{×}×(Z/p_{n}Z)_{×}$, $p$ tends to infinity among the odd prime numbers and $n≥2$ is a fixed integer. This is the analogue of the result obtained in [12] in the prime moduli case.

## Cite this article

Guillaume Ricotta, Emmanuel Royer, Kloosterman paths of prime powers moduli. Comment. Math. Helv. 93 (2018), no. 3, pp. 493–532

DOI 10.4171/CMH/442