Dirichlet generating function for Dirichlet divisor problem

Clear[nn, k, s, m]
nn = 50;
s = 12;
N[Total[Accumulate[DivisorSigma[0, Range[nn]]]/Range[nn]^s], 16]
N[Sum[(Zeta[s - 1] - Sum[HurwitzZeta[s, n/k]*n/k^s, {n, 1, k - 1}])/k, {k, 1, nn}], 16]