Let $p(n)$ be the number of partitions of $n\geq 0$. We can let $n$ be
any complex number in Rademacher's convergent infinite series for
$p(n)$. (See e.g. equation (24) [here](https://mathworld.wolfram.com/PartitionFunctionP.html).)
For what $n$ does it converge? Does it define an analytic function for
such $n$? If so, what are the properties of this analytic function
(singularities, branch points, domain of existence, etc.)? [This question](https://mathoverflow.net/questions/290501)
asked for an analytic continuation of $p(n)$, so I am suggesting a
possible answer.