I often see the use of the integral representation $$\sum_{n < N} \frac{a_n}{n^s} = \frac{1}{2i\pi} \int_{(3)} \sum_{n} \frac{a_n}{n^s} \frac{N^s}{s} ds$$
Is it justified by a kind of application of the residue theorem? Where does this equality comes from?