An integral representation of divisor function. An equation for prime numbers

TL;DR

This paper introduces a novel integral representation of the divisor function using logarithmic residues, offering a new theoretical tool for exploring properties of natural numbers and prime numbers.

Contribution

It proposes a new integral representation of the divisor function via logarithmic residues, potentially advancing number theory research.

Findings

Representation of divisor function using complex analysis

Potential applications for studying properties of natural numbers

A new approach to investigating prime numbers

Abstract

A representation of divisor function $\tau(n)\equiv \sigma_{0}(n)$ by means of logarithmic residue of a function of complex variable is suggested. This representation may be useful theoretical instrument for further investigations of properties of natural numbers.