A Complete Framework and Implementation Path for Research on the Riemann Hypothesis

This paper proposes a comprehensive analytical framework and computational implementation path for studying the Riemann Hypothesis (RH).First,based on anovel rapidly convergent series formula,we construct a strict analytical criterion for the distribution of the zeros of the ζ-function,proving that the Riemann Hypothesis is equivalent to an explicit lower bound estimate on the modulus of a function over the entire right half-plane.Second,we construct a quantum mechanical Hamiltonian corresponding to the spectrum of ζ-function zeros,and through numerical verification,show that its energy level statistics conform to the Gaussian Unitary Ensemble (GUE) distribution, providing a concrete realization of the Hilbert-P´olya conjecture.Third,we extend the framework to higher dimensions,presenting fast algorithms formultiple zeta functions and high-rank automorphic L-functions,along with methods for studying their zero statistics. Finally,we explore potential applications of this theory in cryptography,proposing encryption principles based on L-function zeros and improved primality testing algorithms.This research provides a systematic approach integrating analytic number theory,mathematical physics,and computational science for the eventual proof of the Riemann Hypothesis.