# A question about the behavior of Dirichlet series and its derivatives

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.

• I changed ∑_{n=1}^\infty to \sum_{n=1}^\infty since this alters the positions of the subscripts. Compare $\displaystyle ∑_{n=1}^\infty$ with $\displaystyle\sum_{n=1}^\infty$. ${}\qquad{}$ – Michael Hardy Sep 20 '13 at 19:27