I’ve the next sequence of holomorphic features on $(f_n(s))_{n geq 1}$ on the closed area $R:= { s in mathbb C : Re(s) geq 1}$ by
$$f_n(s) :=
start{circumstances}
log(1+p^{-s})-p^{-s}textual content{, if }n=ptext{ is prime}
0text{, in any other case }
finish{circumstances}$$
(the place we’ve taken the principal department of the logarithm) and I need to present that the sum $sum_{n geq 1} f_n(s)$ is holomorphic on $R$. Now, I learn right here (https://math.stackexchange.com/questions/3675324/weierstrass-theorem-for-closed-domains/3675344#3675344) that Weirstrass’ Theorem would not apply for this closed area $R$, so I don’t know on how I might get began to indicate this outcome. I might be actually grateful for a proof or some hints in that regard. Thanks rather a lot.