Train of thought leading from the zeta function to the Möbius function

(*start Mathematica 8*)
(*Start with Riemann zeta:*)
Zeta[s]
(*Take the logarithm:*)
Log[Zeta[s]]
(*Take the derivative:*)
D[Log[Zeta[s]], s]
Clear[s, c]
(*Generalize it:*)
Limit[Zeta[c] - Zeta[s]*Zeta[c]/Zeta[s + c - 1], c -> 1]
(*See that Zeta[s]*Zeta[c]/Zeta[s+c-1] is the Dirichlet generating \
function of:*)
Table[Limit[
Zeta[s]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(s - 1)]/n^c,
s -> 1], {n, 1, 12}]
(*Which in turn is the Dirichlet generating function of the rows or \
columns of the symmetric matrix:*)
nn = 32;
A = Table[
Table[If[Mod[n, k] == 0, k^ZetaZero[k], 0], {k, 1, nn}], {n, 1,
nn}];
B = Table[
Table[If[Mod[k, n] == 0, MoebiusMu[n]*n^ZetaZero[-n], 0], {k, 1,
nn}], {n, 1, nn}];
MatrixForm[N[A.B]]
(*For comparison,here is a plot of the von Mangoldt function from the \
matrix:*)
ListLinePlot[Total[N[A.B]/Range[nn]]]
(*end*)

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