# Distribution of short sums of classical Kloosterman sums of prime powers moduli

## TL;DR

This paper proves that short sums of normalized classical Kloosterman sums of prime power moduli tend to a Gaussian distribution as the prime tends to infinity, extending previous results to this specific setting.

## Contribution

It establishes the convergence in law of short sums of classical Kloosterman sums of prime power moduli to a Gaussian distribution, under natural conditions.

## Key findings

- Short sums of classical Kloosterman sums of prime power moduli converge to Gaussian distribution.
- Convergence holds as the prime tends to infinity with fixed exponent n ≥ 2.
- Results extend previous work on Kloosterman sums to prime power moduli.

## Abstract

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli $\mathsf{Kl}_{p^n}$, as $p$ tends to infinity among the prime numbers and $n\geq 2$ is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.