Suppose $f:\mathbb{N}_{\geq 1}\to \mathbb{C}$ is an arithmetic function and $$F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s} $$ the Dirichlet series associated to it.

I am trying to prove that if

i) F(s) converges absolutely in $\Re s > c \in \mathbb{R}$.

ii) $ \lim_{s \to c^{+}}F(s) < \infty $

iii) The Taylor expansion of $F$ at $c$, has a positive, say $\epsilon$, radius of convergence

Then $F(s)$ converges in $ \mid s-c \mid <\epsilon $ to its Taylor series.

I am trying to reduce the question to Taylors theorem but I am encountering some conceptual difficulty. Any suggestions or counterexamples will be much appreciated!


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