1. ## Sign uncertainty...

Informations about Diriclet series can be found, among other sources, here...

Dirichlet series - Wikipedia, the free encyclopedia

The 'formal definition' of a Diriclet series is given as...

$\displaystyle f(s)= \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$ (1)

My problem of undestanding is when I try to compute the $\displaystyle f^{'} (*)$ from (1). In the 'manual of mathematic' at my disposal is written...

$\displaystyle \frac{d}{dx} a^{u} = a^{u}\cdot \ln a \cdot \frac{du}{dx}$ (2)

... and setting $\displaystyle a=n$ , $\displaystyle u(x)=-x$ the (2) becomes...

$\displaystyle \frac{d}{dx} n^{-x} = - \ln x\cdot n^{-x}$ (3)

... so that it would be...

$\displaystyle f^{'}(s)= - \sum_{n=1}^{\infty} \frac {a_{n}\cdot \ln n}{n^{s}}$ (4)

My problem is that in Wikipedia is written...

$\displaystyle f^{'}(s)= \sum_{n=1}^{\infty} \frac {a_{n}\cdot \ln n}{n^{s}}$ (5)

The question is easy: which is correct?... (4) or (5)? ...

$\displaystyle \chi$ $\displaystyle \sigma$