Let $z \in D$. Verify that $P ( z ) \neq 0$, that
$$P ( z ) = \lim _ { N \rightarrow + \infty } \prod _ { n = 1 } ^ { N } \frac { 1 } { 1 - z ^ { n } }$$
and that for all real $t > 0$,
$$\ln P \left( e ^ { - t } \right) = - \sum _ { n = 1 } ^ { + \infty } \ln \left( 1 - e ^ { - n t } \right)$$
where $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.