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.)