The Geometric Essence of Riemann Zero-Point Distribution: Based on the PDSM‑NT Primordial Vortex Number Theory

Abstract

The zero-point distribution problem of the Riemann ζ-function serves as the cornerstone of modern analytic number theory, and the Riemann Hypothesis (RH) is a century-old core puzzle spanning mathematics, physics, and cryptography. Classical research frameworks adopt complex analysis as the primary tool, defining the complex variable s = σ+it via analytic continuation. Within the strip region 0 < σ < 1, all non-trivial zero-points are defined as solutions to the equation ζ(s)=0, and the Riemann Hypothesis conjectures that all non-trivial zero-points satisfy σ=1/2. Over the past 160 years, academia has formed three mainstream research paradigms: analytic estimation, large-scale numerical zero-point verification, and random matrix statistical analogy, accumulating massive empirical data and asymptotic conclusions. Nevertheless, the classical framework possesses fundamental academic defects: it only focuses on the algebraic properties and statistical distributions of zero-points while failing to reveal their underlying formation mechanisms. Relying merely on phenomenological fitting and numerical verification, it completely lacks a constructive geometric explanation and cannot essentially clarify the uniqueness and necessity of the Riemann critical line. The PDSM‑NT Primordial Vortex Number Theory proposes a cross-dimensional innovative paradigm, lifting low-dimensional complex-plane number-theoretic problems to the 3-dimensional compact complex CY₃ Calabi–Yau manifold topological system. It completes an ontological transformation of Riemann zero-points from “algebraic roots of functional equations” to “high-dimensional topological vortex annihilation singularities”, fundamentally resolving the essential puzzle of the Riemann Hypothesis. Abandoning the conventional reverse logic of “defining zero-points through functions”, this theory reconstructs a forward topological generation mechanism: the full set of prime numbers is defined as the intrinsic global vortex field source of the CY₃ manifold. Riemann non-trivial zero-points are essentially steady-state annihilation points formed by the complete global topological stress cancellation under the coupling action of multi-prime vortex fields, realizing the deep unification of number-theoretic phenomena and high-dimensional geometric topology.