Transcendental Operational Mathematics: A Unified Iteration Theory of Riemann Zeta Function Family and Riemann Theta Function Family

This paper systematically extends the framework of Transcendental Operational Mathematics to the Riemann zeta function family (including the Riemann zeta function ζ(s), the Hurwitz zeta function ζ(s, a), the Riemann xi function ξ(s), Dirichlet L-functions L(s, χ), the Dedekind zeta function ζK(s), and multiple zeta values) and the Riemann theta function family (including the Riemann theta function θ(z; τ ), the Jacobi theta functions ϑi(z; τ ), and the Siegel theta function Θ(Z; Ω)), together with their inverse operations (the inverse Riemann zeta function ζ−1(w) and the inverse theta function θ−1(w; τ )). The core idea is to extend the iteration count of the translation operators Tv(s) = s + v and Tv,w(z; τ ) = (z + v, τ + w) from natural numbers to complex numbers, utilizing the functional equations of each function to establish a complete theory of fractional-order iterations. We propose an axiomatic system consisting of fourteen independent axioms, prove their consistency and independence, construct fractional-order translation iterations using the functional equations, analyze the singularity structure of complex-order iterations (poles, branch points, natural boundaries), establish a categorical duality between the additive group of numbers and the translation iteration groups, and prove transcendence theorems for fractional-order values (conditional on Schanuel’s conjecture and the Riemann Hypothesis). This paper provides complete numerical algorithms and verification results. All open problems from the original research program have been either completely solved and deeply integrated into the corresponding sections or transformed into well-posed conjectures.