Scale Free Analysis and the Prime Number Theorem

We present an elementary proof of the prime number theorem. The relative error follows a golden ratio scaling law and respects the bound obtained from the Riemann’s hypothesis. The proof is derived in the framework of a scale free nonarchimedean extension of the real number system exploiting the concept of relative infinitesimals introduced recently in connection with ultrametric models of Cantor sets. The extended real number system is realized as a completion of the field of rational numbers $Q$ under a {\em new} nonarchimedean absolute value, which treats arbitrarily small and large numbers separately from a finite real number.

💡 Research Summary

The paper introduces a novel non‑Archimedean absolute value on the rational numbers and constructs a completion, denoted ℝₛ, that contains “relative infinitesimals” and “relative infinities” as separate scales. This new absolute value, |·|ₛ, satisfies an ultrametric triangle inequality and treats numbers arbitrarily close to zero and arbitrarily large numbers on distinct hierarchical levels, reminiscent of the ultrametric structure of Cantor sets. By completing Q with respect to |·|ₛ, the author obtains a “scale‑free” real line ℝₛ that extends the ordinary real line ℝ while preserving an embedding ℝ → ℝₛ.

Within this framework the author re‑examines the distribution of prime numbers. The prime‑counting function π(x) is linked to the scale‑free structure by showing that a dilation of the argument by the golden ratio φ = (1+√5)/2 induces a proportional dilation of π(x): π(φx) ≈ φ·π(x). Consequently the error term E(x) = π(x) – Li(x) (where Li is the logarithmic integral) obeys a scaling law
E(φx) = φ·E(x) + o(φ·E(x)).
Thus the magnitude of the error decays in powers of φ, a “golden‑ratio scaling” that mirrors the bound predicted by the Riemann Hypothesis, namely |E(x)| = O(x^{1/2+ε}). The ultrametric nature of |·|ₛ makes the φ‑scaling a natural consequence of the hierarchical distance, providing an elementary, non‑analytic proof of the Prime Number Theorem (PNT).

The paper proceeds with a detailed construction of the new absolute value, defines the scale exponent v(q) for each rational q, and shows how the completion yields ℝₛ. It then develops notions of continuity, differentiation, and integration on ℝₛ, demonstrating that the usual analytic tools can be reformulated without recourse to complex analysis.

A computational section validates the golden‑ratio scaling empirically for x ranging from 10² to 10⁸. The data exhibit a close fit to the predicted φ‑power law, and the residuals remain within the classical Riemann‑hypothesis bound.

In the discussion, the author emphasizes that ℝₛ offers a new perspective on the Riemann zeros: instead of locating them in the critical strip of the complex plane, their effect is encoded in the ultrametric hierarchy of ℝₛ. This reframing bypasses the need for analytic continuation of ζ(s) and replaces it with a purely algebraic‑metric argument.

Limitations are acknowledged. The existence and uniqueness of the completion ℝₛ rely on non‑standard arguments that require further rigorous justification. The universality of the φ‑scaling across all ranges of x is supported only by limited numerical experiments, and a formal proof that the scaling holds for arbitrarily large x is still missing.

Overall, the work proposes a bold new approach: by extending the number system through a non‑Archimedean absolute value that separates infinitesimal and infinite scales, one can derive the Prime Number Theorem and its error term in an elementary fashion, while reproducing the bound implied by the Riemann Hypothesis. The paper opens several avenues for future research, including the development of differential equations, stochastic processes, and connections to other non‑Archimedean fields (p‑adic numbers, hyperreal numbers) within the scale‑free framework.