Scale Free Analysis and the Prime Number Theorem

TL;DR

This paper introduces a novel elementary proof of the prime number theorem using a scale-free nonarchimedean extension of real numbers, revealing a golden ratio scaling law for the error term and connecting to the Riemann hypothesis.

Contribution

It develops a new framework with nonarchimedean absolute values and relative infinitesimals, providing an innovative approach to prime number distribution analysis.

Findings

Error term follows a golden ratio scaling law

Bound on error respects Riemann hypothesis constraints

New nonarchimedean number system framework

Abstract

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.