Kloosterman paths of prime powers moduli, II

TL;DR

This paper extends previous results on the convergence of Kloosterman sum paths by proving a law of convergence when only the parameter a varies, for prime power moduli with fixed n≥31.

Contribution

It establishes a convergence law for Kloosterman paths over prime power moduli as only a varies, generalizing prior results from prime moduli to prime powers with large fixed n.

Findings

Proves convergence in law of Kloosterman paths for prime power moduli with fixed n≥31.

Extends the analogue of Kowalski and Sawin's prime moduli result to prime power moduli.

Provides a new understanding of the distribution of Kloosterman sums in this setting.

Abstract

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/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 (Z/p^nZ)^\times\times(Z/p^nZ)^\times, 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 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 (Z/p^nZ)^\times, p tends to infinity among the odd prime numbers and n>=31 is a fixed integer.