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Abstract
The purpose of this paper is to investigate one of the functions analogous to the zeta function. Specifically, we introduce and demonstrate properties of the zeta function associated with an abstract Riemann surface having a finite field of constants. The primary achievement will be to establish the equivalence between the Riemann hypothesis in this context and the Hasse-Weil bound for the number of rational points on the mentioned surface (refer to Theorem 6). This expositional paper systematically and rigorously presents the various concepts and fundamental results of the theory without introducing original contributions, all while offering a coherent presentation and an adequate bibliography. In conclusion, an informal illustration is provided on how to approach the Birch–Swinnerton conjecture, one of the renowned "Millennium Problems" (Wiles, 2006).
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