Fractal Number Theory (FNT) and Riemannian Analytic Number Theory

Author: Zhang Suhang

Riemannian analytic number theory established a prime number research system centered on functions of a complex variable, analytic continuation, and the zero spectrum. Its essence is a static, analytic, and statistical mathematical paradigm. By constructing the zeta function through the reciprocals of prime powers, it transforms the problem of prime distribution into the problem of the distribution of zeros in the complex plane, and uses precise analytic formulas to characterize the macroscopic fluctuations and asymptotic laws of primes, thereby constituting the foundational framework of modern number theory.

Traditional number theory has long relied on this analytic path, consistently treating the static formulas of algebra, series, and complex analysis as the sole core tools, and lacking a dynamic description of the generation of prime structures, scale evolution, and topological morphology.

The fractal number theory and prime-fraction operator system established in this study opens up a completely new path for prime number research that is dynamic, topological, and visualizable.

Unlike the Riemannian system, which uses prime reciprocals merely as analytic terms, this theory defines prime reciprocals as independent scale-generating operators. Relying on the FCFG recursive structure, these operators evolve, nest, and fold on the continuous complex plane, forming a fractal topological manifold with self-similar characteristics.

This gives rise to two completely complementary cognitive paradigms for prime numbers:

1. The Riemannian system: an analytic-static paradigm

   Dominated by complex-variable formulas, the zero spectrum, and statistical distributions, it characterizes the macroscopic overall laws of primes, leaning toward numerical values, spectra, and static structures.

2. The fractal number theory system: a topological-dynamic paradigm

   Centered on recursive iteration, scale folding, fractal curvature, and structural generation, it characterizes the microscopic evolutionary rules of primes, leaning toward geometry, dynamics, and visual evolution.

The two systems do not conflict, do not replace each other, and are completely complementary.

Riemann completed the analytic characterization of primes, while this theory completes the topological evolutionary characterization of primes.

The former sees the "statistical laws" of primes; the latter sees the "growth structures" of primes.