Kloosterman paths of prime powers moduli

TL;DR

This paper extends the convergence results of Kloosterman sum paths from prime moduli to prime power moduli, establishing similar distributional limits and convergence in law for the normalized sums as the modulus grows.

Contribution

It generalizes the convergence in distribution of Kloosterman sum paths from prime moduli to prime power moduli, providing new results for fixed exponents n>=2.

Findings

Convergence in law of Kloosterman paths for prime power moduli.

Extension of distributional results from prime to prime power moduli.

Establishment of convergence in the Banach space of continuous functions.

Abstract

Emmanuel Kowalski and William Sawin 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,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, 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^nZ)*.(Z/p^nZ)*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by Emmanuel Kowalski and William Sawin in the prime moduli case.