Multifractal Surface Geometry of Prime Gaps: 3D Visualization and Analysis of Arithmetic Structures

Multifractal Surface Geometry of Prime Gaps: A Novel Approach to the Riemann Hypothesis via Geometric Stability Abstract:This study proposes a new theoretical framework, "Prime Surface Geometry (PSG)," which redefines the distribution of prime numbers—one of the most significant challenges in number theory—through the lenses of differential geometry and statistical physics. While prime gaps (the differences between consecutive primes) have traditionally been treated as one-dimensional statistical objects, we demonstrate for the first time that their underlying structure can be characterized as a "weakly twisted, dome-like surface" in three-dimensional space by employing Multifractal Detrended Fluctuation Analysis (MF-DFA). Our analysis reveals that the surfaces derived from prime gaps possess significantly higher Gaussian and mean curvatures compared to randomized surrogate datasets. Furthermore, these surfaces exhibit a "shape-preserving property," maintaining their geometric integrity across a wide range of scales. Building on these findings, we present the "Prime Multifractal Critical Line (PMCL) Hypothesis," asserting that the central axis of this multifractal surface corresponds mathematically to the critical line (Re(s) = 1/2) of the Riemann zeta function. The observed stability of this central axis provides strong geometric evidence supporting the validity of the Riemann Hypothesis. By integrating quantum chaos theory (GUE statistics) and pure number theory through the universal language of geometry, this research offers a novel, verifiable pathway toward the eventual proof of the Riemann Hypothesis.