A Systematic Theory of Closed-Form Expressions for the Riemann Zeta Function of Real Order

Based on previous research on closed-form expressions for the Riemann zeta function at odd integer arguments, this paper establishes a complete theoretical system for its generalization to real orders. By introducing techniques from analytic continuation of the Gamma function, fractional calculus, and the theory of generalized hypergeometric functions, a unified representation framework for ζ(s) (s ∈ R, s > 1) is systematically constructed. First, the closed-form integral expression for real order and its convergence analysis are rigorously proved. Second, a complete derivation of the representation via hypergeometric functions is established, providing the precise expression for the correction term. Furthermore, the theory of Γ-rational approximation is proposed, with the error estimate being refined and perfected. In particular, explicit closed-form expressions and recursive construction methods are given for the half-integer order case. Finally, fractional differential equations for real order are established, revealing the differential structure of the real-order Zeta function. All theoretical results are equipped with rigorous mathematical proofs, detailed derivations, and numerical verification, providing a systematic methodology and computational tools for the study of the real-order Zeta function.