On a new proof of the Prime Number Theorem

TL;DR

This paper revisits and simplifies a new elementary proof of the prime number theorem within a scale-invariant analysis framework, discussing key concepts and applications, and providing an indirect argument related to the Riemann hypothesis.

Contribution

It offers a more accessible and clearer presentation of a novel elementary proof of the prime number theorem using scale-invariant analysis.

Findings

Simplified proof of the prime number theorem

Discussion of nonarchimedean properties and measure invariance

Implications for the Riemann hypothesis

Abstract

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more simplified manner. Basic properties of some key concepts such as infinitesimals, the associated nonarchimedean absolute values, invariance of measure and cardinality of a compact subset of the real line under an IFS are discussed more thoroughly. Some interesting applications of the formalism in analytic number theory are also presented. The error term as dictated by the Riemann hypothesis also follows naturally thus leading to an indirect proof of the hypothesis.