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where μ(k) is the Möbius function.

Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no x such that M(x) > x. The Mertens conjecture goes even further, stating that there is no x where the absolute value of the Mertens function exceeds the square root of x.

The Möbius function is built-in to Mathematica, the Mertens function is not, but it can be defined with this command:

Mertens[x_] := Plus @@ MoebiusMu[Range[1, x]]

External links

• Values of the Mertens function for the first 50 n are given by SIDN A002321
• Values of the Mertens function for the first 2500 n are given by PrimeFan's Mertens Values Page