Explicit Analytic Expressions for Zeros of the Riemann Zeta Function and Construction of a Theoretical Framework

Based on a rapidly convergent formula for the Riemann zeta function ζ(s), this paper systematically explores explicit analytic expressions for the distribution of its zeros, establishing a unified theoretical framework. The main contributions include: 1) Deriving asymptotic formulas for the zeros based on the Lambert W function and their finite-term closed-form representations; 2) Establishing deep spectral representation connections between zeros and Bernoulli numbers based on operator spectral theory; 3) Constructing integral equations and operator representations for zero distributions; 4) Developing a zero theory for p-adic ζ functions and rigorously correcting their zero assignments through rigid analysis; 5) Proposing efficient numerical verification algorithms and analyzing their complexity; 6) Exploring strict correspondences with quantum chaos and random matrix theory; 7) Providing multiple equivalent forms of the Riemann Hypothesis. This work not only offers new perspectives and tools for understanding zero distributions but also provides new exploratory pathways toward the ultimate proof of the Riemann Hypothesis.