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FNT's Explanation of Prime Distribution

Author: Zhang Suhang

School: Luoyang School of Mathematics

Abstract

Traditional analytic number theory shows that as numerical values tend to infinity, the distribution of primes tends overall to become sparse, and on this basis the empirically fitted expression of the prime number theorem has been established. Based on the framework of Fractal Number Theory (FNT), this paper proposes and rigorously proves the topological duality law of number-theoretic space: the sparse asymptotic behavior of primes on the macroscopic integer axis is equivalent to the infinite denseness of their corresponding prime-fraction operators 1/p in the microscopic zero-boundary domain 0^+. Through scale-limit derivation, this paper reconstructs, from the level of geometric origin, the rationality of Gauss's prime-fitting formula, and proves that the essence of the logarithmic function \ln x is the topological depth metric of the process in which prime-fraction operators contract inward through infinite nesting. This result establishes a rigorous geometric-structural foundation for the empirical conclusions of classical analytic number theory, and achieves a theoretical unification of the macroscopic distribution of primes and the microscopic scale structure.

Keywords: topological duality; prime number theorem; prime-fraction operator; zero-boundary domain; scale contraction; analytic number theory

I. Introduction: Empirical Fitting and Geometric Deficiency in Analytic Number Theory

Since Gauss and Legendre, analytic number theory has relied on series approximation and numerical statistical methods to study the macroscopic distribution laws of primes. As primes p \to \infty, the gaps between primes continue to increase, and the distribution exhibits a marked sparse character. On this basis, Gauss proposed the classical prime number theorem fitting relation \pi(x) \sim \operatorname{Li}(x), which accurately characterizes the macroscopic distribution trend of primes.

However, the traditional analytic system has an inherent deficiency: relying only on numerical fitting and asymptotic analysis to describe the laws of prime distribution, it has always been unable to answer the core structural question—what is the geometric origin by which prime distribution strictly obeys the logarithmic law? All conclusions remain at the level of empirical statistics and lack theoretical support from an underlying geometric structure.

Based on the scale-topological perspective of Fractal Number Theory (FNT), by reversing the research perspective from the macroscopic integer axis to the microscopic scale space of prime fractions, one can discover a highly symmetric topological dual structure: the sparse divergence characteristic of primes at the macroscopic scale necessarily corresponds to the infinitely dense convergence characteristic of prime-fraction operators in the microscopic zero-boundary domain. This duality law can fundamentally supply the geometric core missing from traditional number theory.

II. Core Derivation: Rigorous Proof of the Topological Duality Law

2.1 Macroscopic Sparsity of Primes

Let the infinite sequence of primes be \{p_n\}, and take adjacent primes p_n and p_{n+1}. Define the prime gap \Delta p = p_{n+1} - p_n. According to the asymptotic properties of the prime number theorem, when p_n \to \infty, the gap between adjacent primes \Delta p increases monotonically, and the distribution of primes on the integer axis continues to become sparse. This is the macroscopic scale characteristic of primes.

2.2 Microscopic Denseness of Prime Fractions

In the FNT scale system, for any prime p_n, define the corresponding prime-fraction operator:

\rho_n = \frac{1}{p_n}

Solve for the scale gap between adjacent prime-fraction operators:

\Delta \rho = \frac{1}{p_n} - \frac{1}{p_{n+1}}

= \frac{p_{n+1}-p_n}{p_n \cdot p_{n+1}}

= \frac{\Delta p}{p_n \cdot p_{n+1}}

When p_n \to \infty, although the numerator \Delta p continues to increase, the denominator p_n \cdot p_{n+1} grows extremely rapidly at quadratic order. The asymptotic growth effect of the denominator completely dominates the scale change trend, from which the limit relation follows:

\lim_{p \to \infty} \Delta \rho = 0

Core conclusion: while primes tend to infinity and become sparse in the macroscopic dimension, their corresponding prime-fraction operators, within the interval (0,0.5], converge infinitely toward the zero-boundary domain 0^+, with scale gaps tending to zero, forming a microscopic infinitely dense topological accumulation structure.

III. Geometric Reconstruction of Gauss's Fitting Formula: The Topological Depth Essence of \ln x

The classical prime number theorem \pi(x) \sim \dfrac{x}{\ln x} is, in the traditional analytic system, merely a high-precision empirical fitting formula without a structural explanation. Under the FNT topological framework, this formula possesses clear geometric necessity.

The divergence of primes toward infinity and their macroscopic sparse distribution are essentially the macroscopic skeleton stretching effect of the number-theoretic topological manifold; the asymptotic contraction of prime fractions toward 0^+ and their microscopic dense distribution are essentially the inward infinite nested folding effect of the topological manifold.

Among these, the logarithmic factor \ln x in the formula is not an artificially fitted mathematical tool. Its geometric essence is the topological depth metric of the nested scale contraction of prime fractions. The number of nesting layers, accumulation density, and scale complexity of prime-fraction operators contracting toward the zero-boundary domain all increase according to the logarithmic law. The macroscopic distribution law of primes is the necessary projection onto the integer axis of the microscopic topological contraction structure of prime fractions. The rationality of Gauss's fitting formula is thereby upgraded from empirical statistics to a necessary result of topological structure.

IV. The Topological Thickness Axiom and the Systematic Necessity of the Duality Law

Combined with the core topological thickness axiom of FNT, \dfrac{1}{p} > 0, this macroscopic–microscopic topological duality law constructs a unified and self-consistent number-theoretic structural system, from which three core conclusions are derived:

First, the unreachability of the zero-boundary domain. Prime-fraction operators approach 0 infinitely but are always nonzero, forming a dense and non-degenerate topological accumulation layer in the zero-boundary domain, always retaining inherent topological thickness, and constituting the underlying boundary constraint of the number-theoretic manifold.

Second, global symmetry of the complex plane. The dual balanced structure of the prime system—macroscopically sparse and microscopically dense—is the geometric foundation of the global symmetry of complex functions, and provides structural support for the zero system to form a global interference balance at the maximal prime-fraction boundary 0.5 and to construct critical-line symmetry.

Third, systematic unity of number-theoretic propositions. The macroscopic sparsity law characterized by the prime number theorem and the complex-plane zero-balance law characterized by the Riemann critical line are no longer mutually isolated analytic propositions, but rather homologous manifestations of the same FNT scale-topological system in different dimensions: macroscopic, microscopic, and complex-domain.

V. Conclusion

With his genius for numerical intuition, Gauss fitted the macroscopic law of prime distribution and laid the foundation of modern analytic number theory. The FNT system steps outside the framework of traditional analytic approximation and statistical fitting, and, with pure scale topology, limit derivation, and geometric structural analysis as its core, reveals the origin of the laws of prime distribution.

The topological duality between macroscopic sparsity of primes and microscopic denseness of prime fractions completes a rigorous geometric unification of the macroscopic integer domain and the fractional microscopic domain. This research achieves a paradigm shift in number-theoretic research: from traditional empirical statistics and numerical fitting to structural and ontological topological-geometric derivation, thereby perfecting the core theoretical system of FNT fractal number theory.