Dirichlet g f related to summatory von Mangoldt function

(*This might not be entirely correct!*)

nn = 100;
mm = nn;
s = 10;
cc = 7;
N[a = Table[
    Sum[HurwitzZeta[s, n/k]*n/k^s, {n, 1, k - 1}], {k, 1, nn}], 18];
N[-Sum[Sum[
      If[Mod[n, k] == 0, a[[k]]*MoebiusMu[k], 0], {k, 1, n}]/(n^
       cc), {n, 1, nn}], 18] + Zeta[s - 1] - Zeta[cc]
N[Sum[Sum[
    Limit[Zeta[c]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(c - 1)],
     c -> cc], {n, 1, k}]/k^s, {k, 2, nn}], 18]
1 + %% - %