I was reading a paper in which (at 2.42) it says
$$\sum_{n\leq T}[\frac{1}{n^{1/2+it}} (1-\frac{n}{T})^A] = \zeta(\frac{1}{2}+it) + \mathcal{O}_A(T^{-A/2})$$
I looked online (wikipedia on zeta and on general dirichlet polynomials + google + here) for a proof of this approximation but couldn't find any.
I couldn't prove this myself either: I'm not sure where to start since $\zeta$ cannot be written under this form for $Re(s) < 1$.
Any help appreciated!