Asymptotic Geometrization of Prime Numbers in Gamma Systems and Its Connection to Quantum Information Entropy

This paper introduces and substantiates a novel analytical identity that links discrete prime numbers, the analytic continuation of the factorial (the Gamma function), and the fundamental geometric constant π/36. The convergence dynamics of the series are evaluated within the framework of the Prime Number Theorem (PNT). We demonstrate that the decay rate (asymptotics) of the remainder term is structurally governed by the properties of the Lambert W function and the Euler-Mascheroni constant. In the context of information theory, the formula is interpreted as an expression of the topological capacity of a quantum communication channel, where phase differences of prime numbers determine the mean squared error of the Fisher information matrix. This work opens a new perspective on connecting the non-trivial zeroes of the Riemann zeta function with models of quantum chaos.