Let $\sum\limits_{n=1} ^{\infty} \frac{z^n}{1-z^n}$. Show that

Question

Let $\sum_{n=1} ^{\infty} \dfrac{z^n}{1-z^n}$. Show that

$F(z)=\sum_{n=1} ^{\infty} d(n)z^n$

Where $d(n)$ represents the numbers of divisors of $n$.

Does anyone have any idea how to start?

Answer 1

Hint: for $|z|<1$, $$\frac{1}{1-z^n}=1+z^n+z^{2n}+z^{3n}+\cdots$$ Multiply by $z^n$, sum over $n$, and regroup terms by their exponent.