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

Two elementary formulae and some complicated properties for Mertens function