A New Strategy for Riemann Hypothesis

TL;DR

This paper introduces a novel equation involving Merten's function, offering a promising new approach to the longstanding Riemann Hypothesis by connecting it to properties of square-free integers.

Contribution

It derives a new equation involving Merten's function that provides a fresh perspective and potential technique for addressing the Riemann Hypothesis.

Findings

Derived a new equation involving Merten's function

Proposed a promising technique for Riemann Hypothesis

Established a novel connection between square-free integers and RH

Abstract

A square-free integer is a positive integer that is not divisible by the square of any prime. Merten's function, $M(x)$ is defined as the difference between the number of square free integers with an even number of prime factors and the number of those with an odd number of prime factors up to $x$. It's well known that $M(x)=O(x^{1/2+\epsilon})$ is equivalent to Riemann Hypothesis. In this paper, we derive a new equation involving Merten's function that leads to a promising technique for Riemann Hypothesis.