Bounds of the Mertens Function

TL;DR

This paper explores new properties and potential bounds of the Mertens function, proposing a condition that could lead to a proof of the Riemann Hypothesis based on these bounds.

Contribution

It introduces a likely upper bound for the Mertens function and links this bound to a sufficient condition for proving the Riemann Hypothesis.

Findings

Proposes a likely upper bound for |M(x)| as sqrt(log(x!))

Establishes a sufficient condition for the Riemann Hypothesis based on Mertens function bounds

Discusses properties of the Mertens function related to number theory

Abstract

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient condition to prove the Riemann Hypothesis.