TITLE: The identity $\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})$ 
QUESTION [6 upvotes]: As in the famous Euler product identity, the primes occur on
only one side of the following:
$\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})\ .$
My basic question: Does this identity appear in the literature? 
If not, does the function $\sum_n \ln(n) x^n$?  (It seems 
distantly related to polylogarithms.)  Does it extend beyond the unit disk?
(Computation suggests that behaves quit calmly up to the unit disk - I haven't
detected visible evidence of a natural boundary.)
Is my(?) identity somehow equivalent to the Euler product identity?
Is here some obvious reason why it shouldn't be useful for studying
the distribution of primes?  (For a start it, does prove the infinitude of primes -
if there were only finitely many primes, the coefficients on the left would
have a bounded average.)

REPLY [10 votes]: For every two series $\{a _i\} _{i=1}^{\infty}$ and $\{b _i\} _{i=1}^{\infty}$, which satisfy $a _{n}=\sum _{k|n}b _k$ one has
$$\sum _{n\geq 1}a _n x^n=\sum_ {n\geq 1} b _n \frac{x^n}{1-x^n}$$
this is known as "Lambert transformation". Therefore your identity is equivalent to the familiar statement
$$\log(n)=\sum_{k|n}\Lambda(n)$$
with the von Mangoldt function. This identity is indeed equivalent to unique factorization and therefore to the Euler product identity. As to the literature, this identity is used in the beginning of the proof of the prime number thorem by Delange (1955), for example.