Multifractal Analysis of Prime Gaps: Evidence for Scale-Dependent Geometric Structure

AbstractPrime gaps have long been regarded as irregular fluctuations arising from the distribution of prime numbers. In this work, we investigate whether these fluctuations exhibit persistent multiscale organization using advanced multifractal analysis techniques.We analyze the log-normalized prime-gap sequence \( g_n = (p_{n+1} - p_n)/\log p_n \) with two complementary approaches: the partition function method and Multifractal Detrended Fluctuation Analysis (MF-DFA). Our analysis, conducted on sequences up to $5 \times 10^8$ primes, reveals clear multifractal behavior characterized by nonlinear generalized Hurst exponents \( h(q) \), mass exponents \( \tau(q) \), and a broad singularity spectrum \( f(\alpha) \) with an inverted-parabola shape.Notably, the multifractal spectra exhibit remarkable stability across multiple scales, with key geometric parameters (width \( \Delta\alpha \), peak position, and height) showing consistent behavior. We further interpret the singularity spectrum as a weakly twisted geometric surface in the \( (\alpha, f(\alpha)) \) plane, characterized by moderate local slope variation and consistently negative curvature. These geometric features are absent in shuffled, phase-randomized, and AR(1) surrogate series, supporting their intrinsic origin in the prime distribution.These findings provide strong numerical evidence for a scale-dependent yet robust geometric structure underlying prime-gap fluctuations. We propose "Prime Information Geometry" as a new conceptual framework that bridges number theory, multifractal analysis, and information geometry, opening a potential pathway toward a deeper geometric understanding of prime numbers.Keywords: prime gaps, multifractal analysis, MF-DFA, singularity spectrum, scale invariance, geometric structure, information geometry摘要：素数间隙（prime gaps）长期以来被视作素数分布所引发的不规则涨落。本研究借助先进的多重分形分析技术，探究此类涨落是否具备持久的多尺度组织特性。我们采用两种互补方法对对数归一化素数间隙序列 ( g_n = (p_{n+1} - p_n)/log p_n ) 展开分析：配分函数法与多重分形去趋势波动分析（Multifractal Detrended Fluctuation Analysis，MF-DFA）。针对包含多达 ( 5 imes 10^8 ) 个素数的序列开展的分析结果显示，该序列呈现出清晰的多重分形行为，其特征为非线性广义赫斯特指数 ( h(q) )、质量指数 ( au(q) )，以及呈倒抛物线形态的宽奇异性谱 ( f(alpha) )。值得注意的是，多重分形谱在多个尺度下均表现出显著的稳定性，关键几何参数（宽度 ( Deltaalpha )、峰值位置与峰值高度）均呈现出一致的变化规律。我们进一步将奇异性谱解释为 ( (alpha, f(alpha)) ) 平面上的弱扭曲几何曲面，其特征为适度的局部斜率变化与始终为负的曲率。上述几何特征在洗牌序列、相位随机化序列以及AR(1)替代序列中均未出现，证实了其源于素数分布的内在属性。本研究为素数间隙涨落背后存在尺度依赖却稳健的几何结构提供了强有力的数值证据。我们提出“素数信息几何（Prime Information Geometry）”这一全新概念框架，将数论、多重分形分析与信息几何相联结，为从几何视角深入理解素数的本质开辟了潜在路径。关键词：素数间隙、多重分形分析、MF-DFA、奇异性谱、尺度不变性、几何结构、信息几何