I need estimate the following sum:


$\sum\_{d=1}^{n}\frac{\mu(d)}{d}\sum\_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.


To obtain the main term, I need to find the following sum:


$q+\frac{1}{2}q^2+\cdots+\frac{1}{n}q^n$. Though the second problem looks like calculus problem, I still have no good control about it. 



estimate an sum
Mathematica says that the second sum is equal to:


$$
q^{n+1} (-\Phi (q,1,n+1))-\log (1-q)
$$


What do you want to know about it? This function is also known as the Lerch Transcendent, a lot of info can be fund in [the Wikipedia article.](http://en.wikipedia.org/wiki/Lerch_zeta_function)


