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I am partly repeating myself here. Are the signs of these eigenvalues from this Hermitian matrix "c" equal to the Möbius function? Eigen99 in the Mathematica code is the list of eigenvalues for a 99*99 matrix of the 100*100 matrix that is the discrete Fourier transform of another symmetric matrix t[n, k] that has been mentioned before here on the forum.

Clear[nn, t, n, k, a, b, c, d, e, Mertens, Mobius, Eigen99];
 t[n_, 1] = 1;
 t[1, k_] = 1;
 t[n_, k_] :=
 t[n, k] =
  If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
   t[k - i, n], {i, 1, n - 1}]];
(*nn is size of matrix*)
nn = 100;
e = Range[1, nn];
Do[
m = Table[Table[t[n, k], {k, 1, j}], {n, 1, j}];
a = Fourier[m];
b = Table[Table[a[[n]][[k]], {k, 2, j}], {n, 2, j}];
c = Reverse[b];
(*The Hermitian matrix c*)
MatrixForm[c];
d = Eigenvalues[c];
(*Sums of signs of the eigenvalues*)
e[[j]] = Total[Sign[Round[d, 0.0000001]]] + 1;
, {j, 2, nn}]
Eigen99 = d
Mertens = e
Mobius = Flatten[{1, Differences[e]}]
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