Dirichlet Series Derivative

Question

I tried to obtain the derivative of the Direchlet Series-

$\sum_{n=1}^{\infty} \dfrac{f(n)}{n^s}$

Differentiating each of the terns, I obtained-

$\sum f'(n)n^{-s}+f(n) n^{-s-1}$

However, I should instead get-

$\sum -\dfrac{f(n) \log (n)}{n^s}$

What is incorrect about my approach? Wht should I get the indicated answer?

Answer 1

You calculated $\dfrac{\partial}{\partial n} \dfrac{f(n)}{n^s}$ (which since $f$ is defined only on the natural numbers isn't really meaningful) when you should have calculated $\dfrac{d}{ds} \dfrac{f(n)}{n^s}$.