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I tried DivisorSum[per, DivisorSum[#, d1 = #; MoebiusMu[d1/#] Fij[#] &]/# &] and it actually worked, but I don't understand why it would. I would think that the d1=# should be outside the inner sum, but that does not work.
This code produces a correct result, but gives an error message: Clear[l, CI] 2 CI[{{2, 2}}] + 2 CI[{{1, 2}, {2, 1}}] /. CI[l_] -> j^(Total[l][[2]]) Total::normal: Nonatomic expression expected at position 1 in Total[l]. ...
My paper at https://arxiv.org/abs/1608.08190 has non-Euclidean coordinates for many triangle centers.