Kloosterman paths and the shape of exponential sums

Abstract

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending 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.

27 pages, 3 figures

References in corpus (1)

• A study in sums of products

Cited by in corpus (2)

• On the distribution of the maximum of cubic exponential sums
• On the size of the maximum of incomplete Kloosterman sums