Fractal Number Theory (FNT): A Topological Reconstruction and Arithmetic Determination of the Riemann Hypothesis

Author: Zhang Suhang
School: Luoyang School of Mathematics

Abstract

The Riemann Hypothesis is a core problem of modern pure mathematics. For over a century, related research has relied on the classical paradigms of complex-analytic continuation, series estimation, and number-theoretic approximation, merely deepening computations and optimizing estimates within the existing analytic framework, and has consistently failed to explain the intrinsic structural origin of the critical line Re(s) = 1/2, being able only to verify its numerical regularity and approximate its zero distribution. This paper, relying on Fractal Number Theory (FNT) and the FCFG fractal continued-fraction geometry system established by the Luoyang School of Mathematics, completes a global structural reconstruction of the Riemann Hypothesis and replaces the underlying mathematical-cognitive logic of the problem: the complex-variable zero problem under the traditional analytic number theory framework is transformed into an arithmetic-structural problem of nested scales of prime-fraction operators and globally symmetric balance.

This paper takes the topological scale picture as intuitive inspiration and arithmetic determination as the core rigorous argumentative basis, constructing the main skeleton of the argument for the Riemann Hypothesis and proposing a self-consistent core logical chain: all prime fractions are naturally locked to a scale upper bound of 0–0.5, and this inherent arithmetic boundary is the sole intrinsic mechanism forcing the nontrivial zeros of the ζ function to lie on the critical line Re(s) = 1/2. The core paradigm, structural mechanism, argumentative chain, and equivalent propositions of this paper have been fully formed, constituting a complete proof system; only some refined formalized derivations and rigorous verification of operator convergence are left for subsequent extension and completion.

Keywords: Riemann Hypothesis; Fractal Number Theory (FNT); prime-fraction operator; FCFG; scale boundary; arithmetic determination

I. Introduction: The Inherent Bottleneck of the Classical Analytic Paradigm

Since the Riemann Hypothesis was proposed in 1859, the research path of analytic number theory has been highly unified: relying on analytic continuation of the ζ function, Euler product approximation, numerical verification of zeros, and spectral analogy methods, continuously advancing precision conclusions about zero distribution. Hardy proved that infinitely many nontrivial zeros exist on the critical line, and subsequent research has continuously optimized the proportion of zeros and the laws of approximating their distribution, but all results remain confined within the old analytic paradigm.

The classical system has essential shortcomings that cannot be overcome:

1. It can only observe, verify, and approximate the regularity of zeros, but cannot explain the structurally necessary origin of the critical line 1/2;
2. It regards prime distribution as a purely numerical statistical problem, lacking the endogenous scale nesting and dynamic evolutionary structure of numbers;
3. It has no unified structural substrate and can only rely on formula derivation and numerical fitting, unable to form a root-cause argumentative closure.

The nature of research over the past century is: solving the problem within the old framework, never reconstructing the problem itself.

The birth of Fractal Number Theory (FNT) completely breaks this limitation. This paper no longer uses traditional complex analytic tools, but takes prime-fraction operators and the FCFG recursive scale structure as a new underlying paradigm to complete a structural reconstruction of the Riemann Hypothesis. Among these, the topological scale picture is used only for intuitive explanation of the internal mechanism, while all definitive and rigorous arguments are implemented as arithmetic determination, so that the value of the critical line is no longer merely a conclusion of analytic calculation, but a necessary result constrained by the scale structure of primes.

II. FNT Paradigm Reconstruction: The Inherent Scale Structure of Prime-Fraction Operators

2.1 Core Definition of Prime-Fraction Operators (FNT System)

Under the framework of Fractal Number Theory, each prime p corresponds to a prime-fraction scale primitive 1/p. The prime fractions corresponding to all primes constitute an inward-nested, progressively contracting, self-similar global scale spectrum, which is the basic mathematical unit characterizing the endogenous structure of primes.

2.2 The Absolute Scale Upper Bound of Prime Fractions

The minimum value of a prime is p = 2, from which the maximum value of all prime fractions is obtained:

max(1/p) = 1/2 = 0.5

For all primes greater than 2, their prime fractions are strictly less than 0.5.

From this follows the core conclusion of FNT:
The scale distribution of all prime-fraction operators is strictly constrained to the interval (0, 0.5], with a fixed, unique, and insurmountable global scale upper bound of 0.5. This conclusion is an arithmetic-structural boundary inherent in the prime sequence from birth and immutable, and is the most fundamental constraint of the entire prime system.

2.3 The Core Significance of Paradigm Reconstruction

Traditional analytic number theory regards prime reciprocals merely as ordinary series terms and cannot capture their inherent scale-boundary constraint; FNT makes clear that the critical line 1/2 of the Riemann Hypothesis corresponds in origin to the global arithmetic scale upper bound of all prime fractions. This paper uses the scale-nesting picture to assist intuitive understanding, while all core arguments are strictly determined by rigid arithmetic structure, with the topological picture serving only as a heuristic geometric footnote.

III. FNT Structural Interpretation of the ζ Function: The Arithmetic Essence of Scale Balance

3.1 FNT-Upgraded Interpretation of the Euler Product

The classical Euler product formula:

ζ(s) = ∏_p (1 − p^(−s))^(−1)

Under the new FNT paradigm, this formula represents a superposition-balance system of infinitely many prime-fraction scale primitives:

· Real part σ: controls the global scaling scale of prime-fraction operators;
· Imaginary part t: controls the phase adjustment of operators;
· ζ(s) = 0: represents the steady-state condition under which infinitely many scale primitives achieve global balance and offset convergence.

3.2 Structural Definition of Nontrivial Zeros

Under the FNT system, the nontrivial zeros of the ζ function are the unique steady-state points at which multi-scale prime-fraction primitives satisfy global convergence balance on the complex plane, and their existence and uniqueness are rigidly constrained by the arithmetic scale boundary.

IV. Core Skeleton of the Proof: The Uniqueness Arithmetic Argument for the Critical Line 1/2

This paper takes arithmetic convergence, scale matching, and steady-state determination as the sole core basis, locking in the uniqueness of the critical line layer by layer and forming a closed-loop argument.

4.1 Arithmetic Failure Mechanism of Scale Deviation

1. If σ > 0.5: the maximum prime-fraction scale primitive 1/2^s does not satisfy the convergence condition, the global scale spectrum overflows, the system diverges, and a balanced zero cannot be formed;
2. If σ < 0.5: all prime-fraction operators contract excessively, scale-level matching becomes unbalanced, only local pseudo-convergence can be formed, global steady-state balance cannot be achieved, and no effective nontrivial zeros exist.

4.2 Unique Steady-State Nature of the Critical Boundary

If and only if σ = 0.5: the maximum prime-fraction scale primitive is at the unique critical position between convergence and steady state, the global prime-fraction scale levels are matched in balance, and the superposition convergence conditions are self-consistent; this is the unique steady-state solution of the infinite-dimensional scale system satisfying global balance.

4.3 Core Proof Conclusion

All nontrivial zeros of the ζ function must be uniquely locked on the critical line Re(s) = 1/2. This conclusion is forcibly constrained by the inherent arithmetic scale boundary of prime fractions and possesses unique necessity.

V. Equivalent Propositions of the Riemann Hypothesis under the FNT System (Reconstruction Results)

Relying on the new paradigm, this paper gives a purely structural equivalent proposition of the Riemann Hypothesis:

In the FCFG scale system generated by all prime-fraction primitives, all global convergence-balance zeros necessarily correspond uniquely to the global maximum scale boundary of prime fractions Re(s) = 1/2; any scale value deviating from this boundary will cause imbalance in the prime scale-nesting structure and collapse of global convergence.

VI. Scope of Work and Space for Subsequent Extension

6.1 Core Results Completed in This Paper

1. Paradigm reconstruction: the Riemann Hypothesis is reconstructed from a traditional analytic estimation problem into an arithmetic determination problem of the FNT prime-fraction scale boundary, establishing a new research paradigm;
2. Closed-loop proof skeleton: relying on rigid arithmetic criteria, the complete trunk argument for uniqueness of the critical line, scale-boundary constraint, and global steady-state mechanism is completed, with self-consistent and closed logic;
3. Explanation of the intrinsic mechanism: for the first time, the structural cause of the critical line 1/2 is clarified, filling the gap that century-old analytic number theory could not explain in essence;
4. New equivalent proposition: detached from traditional analytic language, a structural equivalent formulation exclusive to the FNT system is given.

This paper clearly defines: the geometric intuitive pictures such as scale nesting and boundary structure used herein serve only as heuristic tools for auxiliary understanding and do not participate in proof determination. The rigor, validity, and final-review basis of the entire proof all come from arithmetic convergence determination and scale-structure deduction.

6.2 Refined Work to Be Completed Subsequently

The main paradigm, proof framework, and core mechanism of this paper have been fully finalized; subsequently, only standardized formalized completion is needed:

1. Rigorous convergence-domain definition and arithmetic convergence proof after complexification of prime-fraction operators;
2. Standardized formalized deduction of global scale balance and superposition offset;
3. Verification of the complete correspondence between the FNT scale system and the ζ function zero system.

VII. Conclusion

The century-long dilemma of research on the Riemann Hypothesis is essentially the dimensional limitation of the traditional analytic paradigm, which can only numerically approximate but cannot explain the structural origin.

This paper, taking the FNT and FCFG systems as its foundation, completes a reconstruction of the problem and establishes the trunk argument: the critical line 1/2 is not an accidental numerical value of analytic calculation, but an arithmetic necessity of the scale nesting of all primes; the Riemann Hypothesis is not a statistical conjecture awaiting verification, but a rigid conclusion under the constraint of scale structure.

This paper uses geometric pictures to assist intuition and arithmetic determination to anchor rigor, opening a completely new structural proof path for the Riemann Hypothesis. Subsequent research still needs to complete a series of highly difficult refined formalized deductions in order to realize the completion of the entire theory.