Kloosterman paths and the shape of exponential sums

摘要

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums Kl(p)(a), as a varies over F-p(x), and as p tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.