In what follows, for all $z \in D$ we denote
$$P(z) := \exp\left[\sum_{n=1}^{+\infty} L(z^n)\right].$$
Let $z \in D$. Verify that $P(z) \neq 0$, that
$$P(t) = \lim_{N \rightarrow +\infty} \prod_{n=1}^{N} \frac{1}{1-t^n}$$
and that for all real $t > 0$,
$$\ln P(e^{-t}) = -\sum_{n=1}^{+\infty} \ln(1-e^{-nt}).$$