A divided by B and X divided by Y

Clear[f, A, B, n, s, a, b, x, m];
f[x_] = Zeta[x];
A[n_, s_] =
  Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[s + k/n - 1/n], {k, 1, n}];
B[n_, s_] =
  Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[s + k/n], {k, 1, n}];
X[n_, s_] =
  Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[-s + k/n - 1/n], {k, 1,
    n}];
Y[n_, s_] =
  Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[-s + k/n], {k, 1, n}];
n = 120;
s = N[-5 + 2*I, 200];
Block[{$MaxExtraPrecision = 600}, N[A[n, s]/B[n, s], 20]]
Block[{$MaxExtraPrecision = 600}, N[X[n, s]/Y[n, s], 20]]


Clear[xy, s, aa, bb, ab, AB, XY];
aa = 4;
bb = 2;
ab = AB /. Solve[1/(1 - AB) + s == -aa, AB][[1]][[1]];
xy = XY /. Solve[1/(1 - XY) - s == -bb, XY][[1]][[1]];
s = N[-5 + 2*I, 20];
ab
xy