I have the following sequence of holomorphic functions on $(f\_n(s))\_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by 
$$f\_n(s) := 
\begin{cases}
\log(1+p^{-s})-p^{-s}\text{, if }n=p\text{ is prime}\\
0\text{, otherwise }
\end{cases}$$
(where we have taken the principal branch of the logarithm) and I want to show that the sum $\sum\_{n \geq 1} f\_n(s)$ is holomorphic on $R$. Now, I read here (<https://math.stackexchange.com/questions/3675324/weierstrass-theorem-for-closed-domains/3675344#3675344>) that Weirstrass' Theorem doesn't apply for this closed region $R$, so I have no idea on how I could get started to show this result. I would be really grateful for a proof or some hints in that regard. Thanks a lot. 



To show holomorphicity of a certain infinite series of functions
If $\Re(s) > 1/2+\epsilon$ for $\epsilon > 0$ we have $\log(1+p^{-s}) = \sum\_{j=1}^\infty p^{-sj}/j$ so 
$|f\_p(s)| \le \sum\_{j=2}^\infty p^{-\Re(s)j}/j \le p^{-1-2\epsilon}/(1-p^{-1/2-\epsilon})$. Since $\sum\_p p^{-1-2\epsilon}$ converges, your series converges uniformly on $\{s: \Re(s) > 1/2 + \epsilon\}$, and therefore the sum is holomorphic there, and in particular this is true on $R$.


