Dynamics of BCZ Cocycles and the Riemann Hypothesis

Explore the dynamics of BCZ cocycles in this comprehensive lecture from the International Conference on Bounded Sequences 2024. Delve into the BCZ map, a piecewise linear, area-preserving map of a triangle to itself, introduced by F. Boca, C. Cobeli, and A. Zaharescu in 2000. Discover how this dynamical system is used to study statistical properties of Farey fractions and its connection to R.R. Hall's 1970 work on gap distribution. Learn about the map's isomorphism to a Poincare section of the horocycle flow on the modular surface, as revealed by Athreya and Cheung. Understand the link between the BCZ map and characterizations of the Riemann Hypothesis in terms of Farey sequences, established by Franel and Landau in 1924. Gain insights into the dynamical reformulation of the Riemann Hypothesis using BCZ cocycles and explore recent progress made in this area, including findings from Yiming Li's PhD thesis.