Luoyi Fu

This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by extending the Riemann zeta function and Dirichlet $L$ -functions to the elliptic complex domain, based on a newly constructed system of elliptic complex numbers $\mathbb{C}_\lambda(\lambda<0)$. The core challenge addressed is the inherent difficulty in resolving these conjectures within the traditional "circular complex domain" framework ($\lambda=-1$); the author posits that a complete proof is unattainable strictly within this conventional setting. The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers, specifically identifying the limiting case as $\lambda\to 0^{-}$ as the key to the proof. Through rigorous deduction, a bijective correspondence between zeros across different complex planes is established. By employing proof by contradiction and leveraging the correspondence between $\mathbb{C}_\lambda$ (as $\lambda\to 0$) and the circle complex plane $\mathbb{C}$, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven. This paper is organized into three parts: \begin{enumerate} \item Construction and Geometric Properties: The first part details the construction of elliptic complex numbers and their fundamental geometric properties, laying the necessary foundation for subsequent analysis and the proof of the conjectures. \item Analytic Extension: The second part introduces elliptic complex numbers into mathematical analysis, deriving numerous results analogous to those in classical complex variable function theory. \item Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and the Generalized Riemann Hypothesis.