A new analytic continuation for the Riemann zeta function

(*start*)
(*Mathematica*)
n = 10^45;
a = 2;
b = 3;
s = 1/2;
N[(a^(s - 1)*Sum[1/k^(s), {k, 1, a*n}] -
    b^(s - 1) Sum[1/k^(s), {k, 1, b*n}])/(a^(s - 1) - b^(s - 1)), 20]
N[Zeta[s], 20]
(*end*)