# Is the following equality concerning an $L$-function really true?

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.

• as long as the character is not the trivial one, the Dirichlet series of the $L$ function converges to it for $\sigma >0$ (conditionally obviously if $0< \sigma \le 1$) because the partial sums (of the character) are bounded by periodicity and the fact that they sum to zero on a full period, so you can continue doing partial summation for $0<\sigma \le 1$ too – Conrad Jul 18 at 16:00
• Look up the approximate functional equation. – Peter Humphries Jul 19 at 9:47