A Summary of the Riemann Hypothesis and the Siegel Zero Problem under the FNT Paradigm

Author: Zhang Suhang

Affiliation: Luoyang School of Mathematics · Luoyang, Henan

I. Core Positioning of FNT and Prime Fractions

Fractal Number Theory (FNT), relying on the FCFG fractal continued-fraction geometry system, establishes a completely new structural analysis paradigm for prime distribution and complex-variable zero-structure problems. Traditional analytic number theory has long relied on series approximation, inequality estimation, and sieve computation for quantitative research, and can achieve only numerical constraints at the phenomenal level; FNT, by contrast, starts from the endogenous scale structure and inherent arithmetic properties of primes and completes an origin-level, structural paradigm reconstruction of core problems in analytic number theory.

In the FNT system, the prime fraction is the sole basic scale unit characterizing the properties of primes. For all primes p, define the prime fraction:

ρ_p = 1/p

The global range of prime fractions is uniquely locked by the inherent properties of primes:

ρ_p ∈ (0, 0.5]（Zhang Compact Domain）

This range is an arithmetic-structural boundary innate to primes, independent of analytic continuation and function fitting, and is the foundational axiom supporting the entire zero-structure theory. All topological scale pictures in this paper serve only as intuitive heuristic tools; all rigorous determinations and conclusions are implemented in arithmetic-structural deduction.

II. The Dual Core Mechanism of the Prime-Fraction Range (Unified Explanation of Two Major Problems)

The upper and lower bound constraints of the prime-fraction range (0, 0.5]（Zhang Compact Domain ）constitute a unified underlying mechanism for the two core zero problems of analytic number theory; the two sets of conclusions are homologous and self-consistently divided in function:

2.1 Upper-Bound Constraint of Prime Fractions: Locking the Critical Line of the Riemann Hypothesis

The global maximum scale upper bound of prime fractions is 0.5, which is the limiting structural boundary of the prime scale-nesting system.

Under the FNT structural framework, the essence of the zeros of the ζ function is the global equilibrium state after superposition and modulation of infinitely many prime-fraction scale primitives. Only at the symmetry axis Re(s) = 0.5 corresponding to the scale upper bound are the global scale-nesting levels symmetric and the phase conjugations balanced, capable of satisfying the complete cancellation condition of the infinite-dimensional scale system.

From this, a structural determination can be made:

The unique position of the critical line of the Riemann Hypothesis is an arithmetic necessity determined by the maximum scale upper bound of prime fractions.

2.2 Lower-Bound Constraint of Prime Fractions: Disproving the Existence of Siegel Zeros

Prime fractions obey the core arithmetic axiom of FNT:

∀p, 1/p > 0

Any prime fraction is always positive and never becomes zero, always retaining inherent scale thickness.

In the shallow region σ → 1, the scale-nesting levels almost disappear, and the system structure is highly flat. Since all scale primitives are always positive and cannot die out, this region does not possess the structural conditions for global perfect cancellation of infinitely many positive primitives. Even though the Dirichlet character can introduce positive-negative phase modulation, it only changes the rotation direction of the unit and does not change the inherent property that the scale modulus is always positive, and thus cannot compensate for the balance deficiency of the shallow structure.

From this, a structural determination can be made:

Anomalous real zeros near σ = 1 have no structural conditions for continued existence; Siegel zeros do not exist.

III. Core Comparison between the Old and New Paradigms

The research paradigm of traditional analytic number theory has always been confined to the quantitative level: through complex analytic tools, it continuously compresses the zero interval and estimates zero distances, obtaining only restrictive conclusions, and cannot explain the essential cause of zero distribution.

The FNT paradigm of the Luoyang School of Mathematics achieves an essential upgrade:

Relying on the same set of prime-fraction arithmetic structure and the same set of range rules, the upper and lower bounds divide the work to resolve the two major problems:

· Relying on the upper bound 0.5: structurally locking the unique critical position of nontrivial zeros, reconstructing the Riemann Hypothesis;

· Relying on the lower bound > 0: structurally excluding the possibility of existence of anomalous zeros near 1, dissolving the Siegel zero paradox.

IV. Final Conclusion of the School

All deep laws of prime distribution and complex-variable zeros are not numerical phenomena derived from analytic operations, but the external necessary manifestation of the innate scale structure of prime fractions (0, 0.5]（ZCD）.

The FNT paradigm unifies the prime scale system, the complex-plane equilibrium structure, and the mechanism of zero distribution; it uses intuitive pictures to inspire thought and arithmetic determination to finalize conclusions, completing a structural systemic breakthrough that traditional analytic number theory could not achieve, and providing a completely new research framework for prime distribution and L-function zero problems.

— Luoyang School of Mathematics, Zhang Suhang