Mathematica 8

Clear[k, x, n, nn, h]

nn = 90;

g1 = Plot[

N[Total[Table[

Accumulate[Table[MoebiusMu[k], {k, 1, nn}]][[n]]*(x^n/(1 - x^n) -

x^(n + 1)/(1 - x^(n + 1))), {n, 1, nn}]]]

, {x, -5, 5}, PlotRange -> {-5, 5}, AspectRatio -> 1]

g2 = Plot[-x^(-1), {x, -5, 5}, PlotRange -> {-5, 5}, AspectRatio -> 1]

Show[g1, g2]

M(n)=1,0,-1,-1,-2,-1,-2,-2,-2,-1,-2,-2… = oeis sequence A002321.

A002321

which evaluates to:

if

if or .

Singularities at x = -1 and x = 1

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