Fractal Number Theory (FNT): A Structural Reconstruction and Arithmetic Determination of Siegel Zeros

Author: Zhang Suhang

School: Luoyang School of Mathematics

Abstract

The Landau–Siegel zero conjecture is a core problem of analytic number theory. If a real nontrivial zero of a Dirichlet L-function exists whose real part is extremely close to 1, it would directly destroy the uniformity of prime distribution in arithmetic progressions and overturn a large number of classical number-theoretic conclusions. For over a century, traditional research has relied on sieve methods, character sum estimates, and zero-repulsion inequalities, and has been able only to compress the interval of zero existence at the quantitative level, consistently failing to disprove the existence of Siegel zeros from the structural origin.

This paper, relying on the Fractal Number Theory (FNT) system of the Luoyang School of Mathematics, abandons the traditional complex analytic estimation path and takes the axiom of prime-fraction positivity as the sole underlying core, completing a purely structural reconstruction of the Siegel zero problem through a threefold logical framework of "prime fraction greater than 0, prime fraction equal to 0 (hypothetical reductio), prime fraction less than 0."

This paper strictly distinguishes the inherent properties of a single primitive from the overall superposition result of the function, and proves by reductio ad absurdum: the rigid constraint of prime-fraction positivity prohibits global phase cancellation from being completed in the shallow region σ → 1, and Siegel zeros do not possess the structural conditions for existence. The scale picture in this paper serves only as intuitive inspiration; all determinative conclusions are implemented in arithmetic-structural deduction, the trunk logic is fully self-consistent, and refined formalization work is left for subsequent completion.

Keywords: Siegel zero; Fractal Number Theory (FNT); prime-fraction operator; axiom of positivity; structural reductio; arithmetic determination

I. Introduction: From Quantitative Approximation to Qualitative Structure

Classical analytic number theory's research on Siegel zeros has long been confined to the numerical quantitative paradigm: through precise inequalities and sieve estimates, it continuously proves that "Siegel zeros are sufficiently far from the line σ = 1." Such research can only constrain the zero interval, belonging to local phenomenon approximation, and cannot explain the core essence: why the region near σ = 1 naturally cannot generate nontrivial zeros.

The traditional paradigm has always remained at "quantitative constraint" and has never touched "qualitative structure."

Fractal Number Theory (FNT) reconstructs the underlying logic of this problem: it no longer relies on complex analytic estimation, but instead starts from the inherent properties of the scale primitives corresponding to primes, replacing numerical fitting with rigid arithmetic-structural determination, and resolving the existence problem of Siegel zeros at the origin level. All geometric intuitions in this paper are used only to assist understanding, and rigorous arguments are all completed relying on arithmetic structure.

II. Core Axiom of FNT: Positivity of Prime Fractions

In the Fractal Number Theory system, all primes correspond to prime-fraction scale primitives 1/p, which are the basic units constituting the product structure of L-functions and ζ functions.

Core Axiom of FNT Prime Fractions

For any prime p, it always holds that:

1/p > 0

As the prime p → ∞, 1/p approaches 0 infinitely but can never equal 0.

Core distinction of this paper:

The function zero condition L(s, χ) = 0 is the overall result after superposition and modulation of infinitely many primitives, and absolutely does not mean that a single prime-fraction primitive itself can become zero or negative. The inherent property of the primitive is the underlying rigid constraint of all equilibrium conditions of the system.

III. Threefold Logical Framework: Structural Analysis of the Three Value Forms of Prime Fractions

To rigorously clarify the boundary conditions for zero generation, this paper establishes a complete threefold logical system, exhausts all theoretical possibilities, distinguishes real physical properties from hypothetical logical assumptions, and establishes a rigorous foundation for the reductio proof.

3.1 Real State: Prime Fractions Always Greater Than 0

All prime-fraction primitives carry a positive scale, which is the only genuinely existing eigenstate of the FNT system. A single primitive cannot become zero or negative. Global vanishing of the function can only be achieved through phase conjugation and superposition cancellation among different primitives, relying on multi-level scale nesting and symmetric structure.

3.2 Hypothetical State: Prime Fractions Equal to 0 (Counterfactual Assumption for Reductio Only)

If it is assumed that prime fractions can take 1/p = 0, then some scale primitives directly disappear, and the system can achieve overall vanishing without precise phase cancellation. Under this hypothetical premise, zeros existing in the shallow region σ → 1 are permitted.

Key core: This assumption completely conflicts with the FNT axiom of positivity, does not belong to the structural properties of the real prime system, and is used only for logical contrast and reductio.

3.3 Impossible State: Prime Fractions Less Than 0

As scale primitives, prime fractions represent the inherent scale magnitude of primes, and no negative eigenproperty exists. The negative values and phase flips of L-functions all come from the overall system effect of analytic continuation and character modulation, and are not negative values inherent in a single prime-fraction primitive.

(Note: The negative sign introduced by the Dirichlet character χ(p) belongs to phase modulation, which only changes the rotation direction of the primitive on the complex plane and absolutely does not change the underlying property that its scale modulus is always positive.)

IV. Structural Reductio Proof: Siegel Zeros Do Not Exist

4.1 Assumption for Contradiction

Assume that a Landau–Siegel zero exists, that is, there exists a real parameter β → 1⁻ such that:

L(β, χ) = 0

4.2 Derivation of the Core Structural Contradiction

1. Structural characteristics of the shallow region: when β → 1, the scales of all prime fractions undergo almost no nested contraction, the system is in an extremely flat shallow state, and scale-level differences almost disappear.

2. Constraints of the real system: all primitives of the real system strictly satisfy 1/p > 0; no primitive can disappear, and no primitive is naturally negative. To achieve overall vanishing, global multi-level symmetric phase cancellation is necessary.

3. Contrast with the hypothetical system: only by allowing "prime fractions to become zero (1/p = 0)" can a zero be forcibly achieved in the flat shallow region while skipping precise symmetric cancellation.

4. Final contradiction: the premise for the establishment of a Siegel zero is equivalent to tacitly permitting primitives to become zero, which violates the FNT axiom of prime-fraction positivity. In a strictly positive real scale system, the shallow flat structure cannot complete the global perfect cancellation of infinitely many positive primitives.

4.3 Reductio Conclusion

Siegel zeros near σ = 1 do not satisfy the structural equilibrium conditions of the prime-fraction scale system and are structurally impossible to exist.

V. FNT Structural Exclusion Theorem for Siegel Zeros

Theorem: Based on the arithmetic axiom of prime-fraction positivity, the Dirichlet L-function has no real nontrivial zeros near σ = 1.

The hypothetical existence of Siegel zeros depends on the unreal assumption that "scale primitives can become zero"; whereas the real prime scale system is always positive, with no extinction and no negative values, and the shallow region does not possess the structural conditions for global cancellation. Therefore, Siegel zeros are permanently excluded by the inherent structure of the system.

The uniformity of prime distribution in arithmetic progressions is an arithmetic necessity under the constraint of the prime-fraction positivity structure.

VI. Scope of Work and Space for Subsequent Extension

6.1 Core Results Completed in This Paper

1. Paradigm reconstruction: the Siegel zero problem is upgraded from traditional analytic quantitative estimation to a qualitative determination problem of the prime-fraction positivity structure;

2. Completion of threefold logic: through the exhaustive classification of greater than 0, equal to 0 (hypothetical), and less than 0, a rigorous foundation for reductio is established;

3. Filling in the intrinsic mechanism: from the level of the inherent properties of scale primitives, the fundamental reason why Siegel zeros are excluded is explained;

4. Closed-loop trunk argument: the purely arithmetic-structural reductio logic is self-consistent, does not rely on mystical topological argumentation, and the conclusion is rigid and solid.

This paper clearly defines: the scale-structure pictures in the text are only heuristic auxiliary tools and do not participate in proof determination; the final-review basis for all conclusions comes from arithmetic-structural deduction.

6.2 Refined Work to Be Completed Subsequently

1. Standardized formalized deduction of scale superposition imbalance in the shallow region of L-functions;

2. Rigorous supplementary proof of phase cancellation conditions under character modulation;

3. Verification of the complete correspondence between the FNT axiom of positivity and the L-function zero system.

VII. Conclusion

Traditional analytic number theory's research on Siegel zeros has always remained at the numerical constraint at the phenomenal level and cannot touch the structural essence.

This paper, relying on the prime-fraction positivity axiom of the FNT system, through complete threefold logic and structural reductio, directly disproves the existence of Siegel zeros from the origin:

The nonexistence of Siegel zeros is not an approximate conclusion of numerical approximation, but an arithmetic necessity determined by the positivity of all prime scale primitives.

This paper uses intuitive structure to inspire thought and arithmetic determination to finalize conclusions, establishing a completely new structural proof framework for the Siegel zero problem. Subsequent research still needs to complete refined formalized deduction in order to realize the completion of the entire theory.