Let $\psi$ be a Dirichlet character defined mod $q.$ I have seen the claim that for $s = \sigma +it$ fixed and $\sigma >0,$ that $$\sum_{n=1}^y \psi(n)n^{-s} = L(\psi,s) + \underline{O}(y^{-\sigma})$$ with the implicit constant depending on $q$ and $s.$
It is easy to see that this is true, using partial summation, when $\sigma >1,$ but I can not seem to show this otherwise. Is the statement really true if $0 < \sigma <1?$ It seems to me that this can not hold then, but I might be wrong and want to double check.