Let us consider the Dirichlet series: $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ We know that its derivative is given by: $$f'(s)=-\sum_{n=1}^\infty \frac{(\ln n) a_n}{n^s} $$
My question is: What are the necessary and sufficient conditions in which we get that $f$ assumes arbitrarily large and arbitrarily small values and its derivative $f'$ is bounded.