Two elementary formulae and some complicated properties for Mertens function

TL;DR

This paper introduces two simple formulas for the Mertens function, enabling direct computation up to 20 million, and explores its complex properties through extensive numerical analysis.

Contribution

It provides new elementary formulas for Mertens function and applies them to analyze its zeros, extrema, and relation to squarefree integers.

Findings

Calculated M(n) for n up to 20 million.

Identified 16,479 zeros and 10,043 local extrema of M(n).

Explored empirical relations between M(n) and squarefree integers.

Abstract

Two elementary formulae for Mertens function $M(n)$ are obtained. With these formulae, $M(n)$ can be calculated directly and simply, which can be easily implemented by computer. $M (1) \sim M (2 \times 10^7) $ are calculated one by one. Based on these $2\times 10^7$ samples, some of the complicated properties for Mertens function $M(n)$, its 16479 zeros, the 10043 local maximum/minimum between two neighbor zeros, and the relation with the cumulative sum of the squarefree integers, are understood numerically and empirically.