Sieve Procedure for the M\"obius prime-functions, the Infinitude of Primes and the Prime Number Theorem

TL;DR

This paper introduces a sieve-based method to construct M"obius prime-functions, providing new proofs of the Prime Number Theorem and the infinitude of primes through a novel analytical framework.

Contribution

It presents a novel sieve procedure for M"obius prime-functions and offers simple proofs of key number theory results, connecting prime functions with classical theorems.

Findings

Construction of M"obius prime-functions via sieve procedure

Two simple proofs of the Prime Number Theorem

Multiple proofs of the infinitude of primes

Abstract

Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.