An investigation of the asymptotic behavior of the Mertens function

TL;DR

This paper refutes the earlier belief that the Mertens function behaves like a simple random walk obeying the law of the iterated logarithm, providing a refined asymptotic estimate instead.

Contribution

It demonstrates why the law of the iterated logarithm does not apply to the Mertens function and establishes a new refined asymptotic estimate.

Findings

The law of the iterated logarithm does not hold for the Mertens function.

A new refined asymptotic estimate for the Mertens function is proved.

The paper clarifies the asymptotic behavior of the Mertens function.

Abstract

It was believed that the Mertens function is a simple random walk in the first versions of the article, so its asymptotic behavior obeys the law of the iterated logarithm. In the latest version of the article we show why the asymptotic estimate -- the law of the iterated logarithm -- does not hold for the Mertens function. A refined asymptotic estimate for the Mertens function is proved.