Fundamental theorem of arithmetic, prove that these matrices are the same apart from the main diagonal.

Question

I know and I can prove that the fundamental theorem of arithmetic is encoded uniquely by this recurrence written in Mathematica 8:

(*start*)
Print["Log[n] is to:"]
Clear[t, s, n, k, z];
z = 1;
nn = 12;
t[n_, k_] := 
  t[n, k] = 
   If[k == 1, 
    Log[n] - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], 
    If[Mod[n, k] == 0, t[n/k, 1], 0], 0];
MatrixForm[
 Table[Table[Limit[t[n, k], s -> z], {k, 1, nn}], {n, 1, nn}]]
 (*end*)

Prove that by replacing $\log(n)$ with $\zeta(s)$ (Riemann zeta)

(*start*)
Print["as Zeta[s] is to:"]
Clear[t, s, n, k, z];
z = 1;
nn = 12;
t[n_, k_] := 
  t[n, k] = 
   If[k == 1, 
    Zeta[s] - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], 
    If[Mod[n, k] == 0, t[n/k, 1], 0], 0];
MatrixForm[
 Table[Table[Limit[t[n, k], s -> z], {k, 1, nn}], {n, 1, nn}]]
(*end*)

we get the same matrix apart from the main diagonal $n=k$ where the entries are $\infty$.

Edit 14.3.2016: A good point to start is probably: $$\log(n)= \lim_{s\to 1} \, \zeta (s)\left(1-\frac{1}{n^{s-1}}\right)$$