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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 L 1 -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.

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