# To show holomorphicity of a certain infinite series of functions

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.

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 $$R$$.