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Fractal Number Theory (FNT): Topological Reconstruction and Proof of Siegel Zeros

Author: Zhang Suhang

School: Luoyang School of Mathematics

Abstract

The Landau–Siegel zero conjecture is a core problem in analytic number theory. If a Dirichlet L-function has a real zero extremely close to 1, it will lead to an imbalance in the distribution of primes. Based on the framework of Fractal Number Theory (FNT), this paper completely breaks away from the cumbersome path of traditional analytic estimates and, with the "prime-fraction topological axiom" as its core, completes a topological reconstruction and proof of this problem. This paper points out that primes tend to infinity and prime fractions tend to infinitesimal, but are absolutely greater than 0. This topological thickness axiom determines that interference cancellation in the complex manifold can be achieved only at the maximum-scale boundary (0.5). Any hypothesis attempting to produce a zero (=0) at the edge (near 1) will inevitably cause manifold tearing due to violation of the conservation of topological tension. Through a purely topological proof by contradiction, this paper proves the nonexistence of Siegel zeros at the geometric origin.

Keywords: Siegel zero; Fractal Number Theory (FNT); prime-fraction operator; topological thickness axiom; interference cancellation

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I. Introduction: From Analytic Estimates to Topological Qualitativeness

Traditional analytic number theory (including Mr. Yitang Zhang's latest breakthroughs) is devoted to using extremely complex sieve methods and zero-repulsion methods to prove how far a Siegel zero \beta is from 1. But this has always remained at the level of "quantity."

The intervention of the FNT system directly elevates "quantity" to "quality": in the structure of complex topological space, a Siegel zero simply cannot obtain a legitimate geometric coordinate.

II. Core Axiom of FNT: The Topological Thickness of Prime Fractions

In the FNT framework, a prime p corresponds to the inward-collapsing prime-fraction operator 1/p. The essence of \zeta(s)=0 or L(s,\chi)=0 is the global interference cancellation (function value returning to zero) that occurs after the superposition of infinitely many prime-fraction operators.

For this reason, we introduce the most fundamental foundation of FNT:

The Prime-Fraction Topological Thickness Axiom:

As the prime p \to \infty, the prime fraction 1/p \to 0. But no matter how deeply it contracts, the spatial tension of the prime-fraction operator never disappears; that is, 1/p > 0 always holds.

That is: 0 is the topological horizon that prime fractions can never reach. The achievement of any "return to zero (=0)" is not the disappearance of operators, but the perfect symmetric cancellation achieved by operators through a specific geometric posture.

III. Analysis of Core Cases: The Topological Essence of =0 and <0

According to the axiom that prime fractions are always greater than 0, we directly give a topological qualitative analysis of the zeros (=0) and negative divergence (<0) of the L-function:

3.1 Case One: The Topological Meaning of <0 (Topological Flip)

On the negative half-axis of analytic continuation, the prime-fraction operator 1/p^\sigma (when \sigma<0) becomes p^{|\sigma|}, which expands outward rapidly. In FNT space, this represents a topological flip, from inward collapse to outward explosive expansion. At this time, the manifold completely loses centripetal force and cannot produce conjugate cancellation; there exist only topological singularities (trivial zeros), and there is no Siegel zero as discussed here.

3.2 Case Two: The Topological Meaning of =0 (Tension Cancellation)

To make infinitely many prime-fraction operators that are >0 cancel to 0, the system must achieve an absolute mirror-symmetric balance. The only balance axis that can provide such "perfect nesting and conjugate cancellation" is firmly locked by the maximal prime-fraction primitive 1/2 at the critical line \sigma = 0.5.

IV. Reductio ad Absurdum: Why Can a Siegel Zero Not Approach 1?

The essence of a Siegel zero is an attempt to realize =0 in a region where the real part is extremely close to 1.

Contradiction hypothesis: Suppose there exists a Siegel zero \beta \to 1 such that L(\beta, \chi) = 0.

Geometric contradiction:

1. When \beta \to 1, the prime-fraction operators almost stop inward nesting, and all are squeezed into an extremely flattened shallow surface.

2. In this "flat state," although the phases between operators can be modulated by the character \chi, they still carry an irremovable topological thickness (1/p > 0).

3. To forcibly cancel these >0 tensions to 0 at a flat edge lacking deep nesting, the only way is to produce local extreme stretching. This necessarily causes the local curvature to blow up and triggers topological tearing.

4. According to the law of manifold conservation, for self-preservation the system will generate a strong geometric repulsion, forcibly kicking out any coordinate that attempts to "return to zero" near 1.

Derived conclusion: A Siegel zero is a geometrically unstable distortion point and cannot persist in a region near 1.

V. Ultimate Theorem and Conclusion

FNT Siegel Zero Repulsion Theorem:

Because prime fractions are always greater than 0, interference cancellation in the complex manifold can achieve absolute mirror balance only at the maximum-scale boundary (0.5). Any hypothesis that deviates from this axis and attempts to return to zero at the boundary near 1 violates the prime-fraction topological thickness axiom, and will inevitably be torn apart and repelled by the system's topological tension. Therefore, Siegel zeros do not exist, and the uniformity of prime distribution is a geometric necessity of the tension of the topological manifold.

VI. Conclusion

Previous researchers, relying on precise analytic inequalities and sieve tools, made rigorous quantitative constraints on the range of Siegel zeros; this paper, relying on the FNT prime-fraction topological axioms, directly disproves the existence of Siegel zeros from the level of geometric origin.