Explicit Prime Distribution Theory Based on a Rapidly Convergent Framework: From Zeros of the Riemann Zeta Function to the Hardy–Littlewood Conjectures

This paper establishes a complete and explicit theory of prime distribution based on a rapidly convergent summation formula for the Riemann zeta function ζ(s). We first rigorously derive and prove an explicit analytic formula for the non-trivial zeros ρ of ζ(s), expressing their imaginary parts as rapidly convergent series involving Bernoulli numbers, the Gamma function, and values of ζ(2k). This breakthrough allows us to obtain completely explicit, rapidly convergent series representations for the prime counting function π(x) and the twin prime counting function π2(x), entirely eliminating the need for the implicit summation over zeros required in classical explicit formulas. We then systematically generalize this theory to arbitrary prime k-tuples, establishing a complete explicit formula system for the Hardy–Littlewood conjectures, and providing a universal rapidly convergent expression for the counting function π(x;H). Furthermore, we present an explicit analytic formula for the Hardy–Littlewood constants (singular series) S(H), expressing them as absolutely convergent series involving the prime zeta function and combinatorial parameters of the tuple, accompanied by rigorous proof. Our theoretical framework not only provides novel tools for studying prime distribution but also offers new perspectives on several unsolved problems in analytic number theory. Additionally, we systematically optimize all significant constants within the system, presenting best possible values or directions for optimization, thereby enhancing the precision and practical utility of the theory.