Toward a Proof of the Riemann Hypothesis: Insights from Elliptic Complex Analysis

This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by ex-tending the Riemann zeta function and Dirichlet L -functions to the elliptic complex domain, based ona newly constructed system of elliptic complex numbers Cλ(λ < 0) . The core challenge addressed is theinherent difficulty in resolving these conjectures within the traditional ”circular complex domain” frame-work (λ = −1); the author posits that a complete proof is unattainable strictly within this conventionalsetting.The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers,specifically identifying the limiting case as λ → 0− as the key to the proof. Through rigorous deduction,a bijective correspondence between zeros across different complex planes is established. By employingproof by contradiction and leveraging the correspondence between Cλ (as λ → 0) and the circle complexplane C, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven. Thispaper is organized into three parts:(1) Construction and Geometric Properties: The first part details the construction of elliptic complexnumbers and their fundamental geometric properties, laying the necessary foundation for subsequentanalysis and the proof of the conjectures.(2) Analytic Extension: The second part introduces elliptic complex numbers into mathematical anal-ysis, deriving numerous results analogous to those in classical complex variable function theory.(3) Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and theGeneralized Riemann Hypothesis.