Let $\zeta$ denote the Riemann zeta function and $T \in \mathbb{R}$. What is the residue of $\log \zeta(s + iT)$ at $s=1-iT$ ?
My attempt: Since $\zeta(z)$ has a simple pole at $z=1$ with residue $1$, note that Res $_{s=1-iT} \log \zeta (s + iT) = \log $ Res$_{s=1} \log \zeta(s)$ =$\log 1=0$ ?
ADDENDUM: Below are replies to the comments, somehow I can't post a comment.
@Conrad: Yes, $\log \zeta(s+iT)$ is not continuously defined around $s=1-iT$. However, making a detour around the point $s=1-iT$ produces some integral over some circular contour, which can be evaluated as the residue at that point.
PS: This question is a reformulation of On the proof of Theorem 13.8 of Titchmarsh was said to be unclear.
@Greg Martin, thanks. Can you please kindly answer this On the proof of Theorem 13.8 of Titchmarsh ?