To show holomorphicity of a certain infinite series of functions

Question

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.

Answer 1

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 $R$.