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.



Does Rademacher's convergent series for p(n) define an analytic function?
Edit. We can write the series in the form
$$p(n)=\sum A\_k(z)\frac{d}{dz}f(z/k^2),$$
where $|A\_k(z)|\leq Ck^{1/2}e^{C\_1(\Im z)^+},$ where $y^+=\max\{ y,0\},$ and $C\_j$ are various positive absolute contants, $f(z)=(\sinh\sqrt{z})/\sqrt{z},\; z=C\_2(n-1/24)$.
Notice that $f$ is an even entire function of order $1/2$, normal type so $f'(0)=0$. So
$$p(n)=\sum c\_k(z)f'(z/k^2),$$
where $|c\_k(z)|\leq C\_3k^{-3/2}e^{C\_1(\Im z)^+}$. Since $f'$ is an entire function of order $1/2$, normal type and $f'(0)=0$, we have an estimate of the form
$$|f'(z)|\leq C\_2|z|e^{A\sqrt{|z|}},$$
and we see that the sum above must satisfy the estimate
$$|p(z)|\leq C\_4e^{C\_1(\Im z)^++A\sqrt{|z|}}.$$


Therefore $p(n)$ extends to an entire function in the plane of exponential, type. An interesting question about such a function is the location of its zeros,
and the related question, asymptotics for large $|z|$. One could try to plot $|p(z)|$ on a computer and look what to expect. Even a plot on the imaginary line may tell us something interesting.


Based on this argument, and pictures of Fredrik Johansson, I conjecture that zeros are asymptotic to some parabola $y=C\sqrt{x}$ in the right half-plane, and close to the real axis in the left half-plane (the pictures indicate infinitely many negative zeros).
To prove this one has to prove the asymptotics
$$p(z)=\exp\left(B(\Im z)^++A\Re\sqrt{z}+o(\sqrt{|z|})\right)$$
outside some small circles around zeros.


Remark. Unfortunately, this crude argument gives $C\_1=B=2\pi$, and in such class of entire functions extension from integers to the complex plane is not unique.


