Mertens function Lambert series

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

y = M(n)(\frac{x^{n}}{(1 - x^{n})} -  \frac{x^{(n + 1)}}{(1 - x^{(n + 1)})})

which evaluates to:

y = x if -1 \le x \le 1

y = -\frac{1}{x} if x \textless -1 or x \textgreater 1.

Singularities at x = -1 and x = 1

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