On the properties of generalized harmonic and oscillatory numbers. Simple proof of the Prime Number Theorem

TL;DR

This paper introduces sum identities for generalized harmonic and oscillatory numbers, facilitating a sieve-based approach to analyze their properties and providing a straightforward proof of the Prime Number Theorem.

Contribution

It presents new sum identities for these numbers and offers a simple, alternative proof of the Prime Number Theorem using these identities.

Findings

Sum identities enable better understanding of number properties.

Asymptotic behavior of these numbers is clarified.

A simple proof of the Prime Number Theorem is achieved.

Abstract

We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their asymptotic behavior. On the basis of these identities a simple proof of the Prime Number Theorem is represented.