# Kloosterman paths of prime powers moduli, II

Guillaume Ricotta, Emmanuel Royer & Igor Shparlinski

avril, 2020

### Résumé

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S\left(a,b;p^n\right)/p^{n/2}$ converge in law in the Banach space of complex-valued continuous function on $[0,1]$ to an explicit random Fourier series as $(a,b)$ varies over $\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times\times\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times$, $p$ tends to infinity among the odd prime numbers and $n\geq 2$ is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case. The purpose of this work is to prove a convergence law in this Banach space as only $a$ varies over $\left(\mathbb{Z}/p^n\mathbb{Z}\right)^\times$, $p$ tends to infinity among the odd prime numbers and $n\geq 31$ is a fixed integer.

Publication

*Bull. Soc. Math. France*