  We investigate the properties of the BCZ map. Based on our findings, we
define the moduli space associated with its excursions. Subsequently, we
utilize the framework we build to establish a discretized analog of the Riemann
hypothesis (RH) that holds in a stronger sense from a dynamical perspective.
The analog is founded upon a reformulation of the RH, specifically in terms of
estimates of L1-averages of BCZ cocycle along periodic orbits of the BCZ map.
The primary tool we will rely on is the generalized arithmetic sequence, which
we will define and discuss.
