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