Kloosterman paths of prime powers moduli

In [12], the authors 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, b(0) ; p)/p(1/2) converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z / p(Z))(x), b(0) is fixed in (Z / p(Z))(x) and p tends to infinity among the odd prime numbers. This article considers the case of S(a; b(0) ; p(n/2), as a varies over. (Z = p(n)Z)(x) , b(0) is fixed in (Z / p(n)Z)(x), 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(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 in [12] in the prime moduli case.