2018/10/02 by Ricotta, Guillaume, Royer, Emmanuel, Shparlinski, Igor
#11L05 #11T23 #60F17 #60G17 #60G50 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1810.01150
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;pn)/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/pnZ)^××(Z/pnZ)^×, 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/pnZ)^×, p tends to infinity among the odd prime numbers and n>=31 is a fixed integer.