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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.

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

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