Kloosterman paths of prime powers moduli

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 $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$, $b_0$ is fixed in $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$ 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 $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, $b_0$ is fixed in $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, $p$ tends to infinity among the odd prime numbers and $n\ge 2$ 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 $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }\times (\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, $p$ tends to infinity among the odd prime numbers and $n\ge 2$ is a fixed integer. This is the analogue of the result obtained in [12] in the prime moduli case.