Another estimating the absolute value of Mertens function

TL;DR

This paper proposes an inversion-based estimation method for the absolute value of the Mertens function, suggesting it can be approximated by a specific formula involving x and a small parameter epsilon.

Contribution

It introduces a novel estimation approach for |M(x)| using an inversion technique and a formula involving epsilon and x, providing bounds for large x.

Findings

Proposes a formula for |M(x)| involving epsilon and x.

Shows that for large x, |M(x)| can be bounded by the proposed estimate.

Provides a method to choose epsilon for bounds on |M(x)|.

Abstract

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$ (where $x$ is an appropriately large real number, and $\varepsilon$ ($0<\varepsilon<1$) is a small real number which makes $2x+\varepsilon$ to be an integer). For any large $x$, we can always find an $\varepsilon$, so that $\vert M(x) \vert < \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$.