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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.

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  • $\begingroup$ 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 $\endgroup$ – Conrad Jul 18 at 16:00
  • $\begingroup$ Look up the approximate functional equation. $\endgroup$ – Peter Humphries Jul 19 at 9:47

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