Operational Mathematics of Series: Extending the Iteration Count to the Complex Domain

This paper systematically establishes the operational mathematics of series, a new branch of mathematics that treats series as infinite iterations of addition. The core idea is to extend the number of summands in a series from natural numbers to integers, rational numbers, real numbers, and complex numbers, thereby unifying finite sums (partial sums) and infinite sums (series) within a single analytic framework. We propose a complete axiomatic system, rigorously define integer-order, fractional-order, real-order, and complex-order partial sums, and prove the existence of analytic continuations of the partial-sum function using Mellin transforms, Euler--Maclaurin summation, and a systematic analysis of singularities. Uniqueness theorems under natural growth conditions are established. A fundamental structural discovery is the classification of divergent series summation methods (Abel, Ces\`aro, Borel, zeta regularization) as limits of the analytically continued partial-sum function along different paths to infinity. We construct a Hilbert--P\'olya self-adjoint operator whose spectrum is directly related to the zeros of a corrected series zeta function, yielding an unconditional proof that all non-trivial zeros of this corrected function lie on the critical line (z) =1/2. A conditional reduction of the classical Riemann Hypothesis to a concrete series identity is obtained. The paper fully integrates the essential features of series: rearrangement, Cauchy product, conditional convergence, and summation methods. Connections to quantum field theory (zeta regularization, renormalization group), turbulence theory, and information theory are explored. Numerical algorithms with exponential convergence and rigorous error bounds are provided. The theory is self-contained, and every essential statement is accompanied by a detailed proof.