Kloosterman paths of prime powers moduli, ii

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(n)Z)(x) x (Z/p(n)Z)(x), 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(n)Z)(x), p tends to infinity among the odd prime numbers and n >= 31 is a fixed integer.