  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.
