I was reading A Course in Arithmetic by Serre and on page 74 he defined the log of the Dirichlet L functions.
(Previously, he showed that $L(1,\chi)\neq 0$ when $\chi\neq 1$).
Based on that fact he showed that $log(L(1,\chi))$ is bounded as $s\rightarrow 1$. That is what I don't get. If he was using the continuity of the log then shouldn't he show that $L(1,\chi)\notin \mathbb{R_-}$ (if he is using the principal logarithm)
Can someone care to explain how he concluded that $\log(L(1,\chi))$ is bounded as $s\rightarrow 1$?
$\DeclareMathOperator{\Log}{Log}$ Let us denote the principal branch of the logarithm by $\Log$. Then $\log L(s,\chi)$ is for $\operatorname{Re} s > 1$ defined as $$\sum_p -\Log\bigl(1 - \chi(p)p^{-s}\bigr)\,.$$ (Every other branch of $\log L(s,\chi)$ differs from this one by a multiple of $2\pi i$.)
Then there is a $\sigma_0 \in (1,2)$ such that $\log L(s,\chi) = \Log \bigl(L(s,\chi)\bigr)$ holds for all $s$ with $\operatorname{Re} s > \sigma_0$, but it does not follow that $\log L(s,\chi) = \Log \bigl(L(s,\chi)\bigr)$ holds for all $s$ in the half-plane $\operatorname{Re} s > 1$ such that $L(s,\chi)$ belongs to the domain of $\Log$, i.e. $\mathbb{C}\setminus (-\infty,0]$. This happens if and only if $L(s,\chi)$ never attains a negative real value in that half-plane.
But this isn't needed for the conclusion that $\log L(s,\chi)$ remains bounded in a neighbourhood of $1$. Indeed, since $L(1,\chi) \neq 0$ there is an $\varepsilon > 0$ such that $L(s,\chi) \neq 0$ for all $s$ with $\lvert s - 1\rvert < \varepsilon$. On that disk, since it is simply connected, there is a holomorphic function $f$ with $L(s,\chi) = e^{f(s)}$. Now we also have $L(s,\chi) = e^{\log L(s,\chi)}$ on the half-disk $U = \{s : \lvert s-1\rvert < \varepsilon, \operatorname{Re} s > 1\}$, thus there is an integer $k$ with $\log L(s,chi) = f(s) + 2\pi i k$ for $s \in U$. Then $$g(s) = \begin{cases}\log L(s,\chi) &\text{if } \operatorname{Re} s > 1 \\ f(s) + 2\pi i k &\text{if } \lvert s - 1\rvert < \varepsilon \end{cases}$$ is a holomorphic branch of the logarithm of $L(s,\chi)$ on $\{s : \operatorname{Re} s > 1\} \cup \{ s : \lvert s-1\rvert < \varepsilon\}$. As a function that is holomorphic on a neighbourhood of $1$ it is bounded on some neighbourhood of $1$.
Actually, this shows that $\lim_{s \to 1} \log L(s,\chi)$ exists, not only the boundedness.