For the one that knows about Zeta

Quick question.

In the critical band where we can use the Dirichlet sum, how fast does this serie converge to zeta? If we sum until N, is it possible to give an equivalent of the rest of the sum (that should depend on s) or not? I tried numerically, I have the impression that absolute value of the sum converges to Zeta as |sum[1...N]-Zeta(s)|=ln(n)/n^a (where s=a+ib).

Is it correct? If yes, how could we prove it?

I realize it is not a easy question and I guess there is unfortunately no answer to that...(Crying)