Kloosterman paths of prime powers moduli, II
Number Theory
2019-05-08 v3
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.
Cite
@article{arxiv.1810.01150,
title = {Kloosterman paths of prime powers moduli, II},
author = {Guillaume Ricotta and Emmanuel Royer and Igor Shparlinski},
journal= {arXiv preprint arXiv:1810.01150},
year = {2019}
}