Kloosterman sums are fundamental objects of analytic number theory, appearing, for example, as Fourier coefficients of Poincaré series or in the circle method. A detailed understanding of the properties of these sums has become an area of interest in itself.
These sums are parameterized by an integer: the modulus. They are constructed by successive additions of complex numbers of norm 1, and can be considered as a sequence of partial sums. Plotted in the complex plane, these partial sums lead to some very intriguing figures, known as Kloosterman paths (see, for example https://blogs.ethz.ch/kowalski/the-kloostermania-page/).
Kowalski & Sawin have developed a probabilistic interpretation of the distribution of these paths in the case of the prime modulus. I shall present an interpretation in the case where the modules are powers of prime numbers. This work was carried out in collaboration with Guillaume Ricotta on the one hand, and Guilaume Ricotta and Igor Shparlinski on the other.