# Convergence of Rademacher's formula: Extending the partition numbers to complex index

$$p(n) = \frac{1}{\pi \sqrt{2}} \sum_{k=1}^{\infty} \sqrt{k}\ A_k(n)\ F_k'(n)$$ $$A_k(n) = \sum_{0 \le m < k, \gcd(m, k) = 1} e^{i\pi\left(s(m, k) - 2nm/k\right)}$$ $$F_k(x) = \frac{1}{\sqrt{x - \frac{1}{24}}} \sinh\left(\frac{\pi}{k} \sqrt{\frac{2}{3}\left(x - \frac{1}{24}\right)}\right)$$

and $s(m, k)$ is the Dedekind sum given by

$$s(m, k) = \sum_{n=1}^{k} \left(\left(\frac{n}{k}\right)\right)\left(\left(\frac{mn}{k}\right)\right)$$

where $((x))$ is the sawtooth function

$$((x)) = \begin{cases} x - \lfloor x \rfloor - \frac{1}{2}, &\mbox{if } x \in \mathbb{R} \setminus \mathbb{Z}\\ 0, &\mbox{if }x \in \mathbb{Z} \end{cases}$$.

$p(n)$ is the $n$th partition number. But here's the interesting thing: this formula also actually seems to work not just for natural, but real and complex values of $n$ as well! So we could perhaps think of Rademacher's formula as giving an extension of the partition-number function to real and complex indices, much as how the gamma function extends the factorial. Albeit, however, it is complex-valued at real indices.

But the question I have is, where does this converge, when the range of $n$ is expanded from $\mathbb{N}$ to $\mathbb{C}$? It seems to converge okay for real values of $n$, and also those for complex $n$ (perhaps should now be called $z$?) with negative imaginary part. But what about with positive imaginary part? The formula is slow, and numerical experiments with $n = 2i$ don't seem to help. It looks like it diverges, but I'm not totally sure. This is a very complicated series formula, and I'm not sure where one would even begin to analyze it to determine the region of convergence.