Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here.
After Ramanujan and Hardy found the infinite sum representation of the partition function $p(n)$, Rademacher went about simplifying their proof; the form generally seen involves integrating $\frac{P(q)}{q^{n+1}}$ along a circle centered at $0$, where $P(q)=\sum_{n \geq 0} p(n) q^n$. We fix an integer $N$ and consider the Farey sequence of fractions with denominator at most $N$; we then split the circle into parts, "centering" each part at a given element of this Farey sequence; as $N$ goes to infinity (and our radius goes to $1$), this gives us the infinite sum we want.
Later, Rademacher rewrote this proof using a different contour along the upper half plane (which can be sent to the unit disc by $z \mapsto e^{2\pi i z}$). He integrated along the Ford circles, starting at $i$ and going to $i+1$ with a sequence of paths, each path going further 'down' the Ford circles than before; he constructed the ford circles for the $N$th Farey sequence, started at the top of the largest circle, followed an arc until he reached the last tangency with another circle, followed that circle to the right, etc. This ultimately leads to a cleaner proof, I'd guesstimate about a third the length of the original. However...
It's not clear at all how he thought to use this method. Why did Rademacher choose to integrate along the Ford circles? Is it just because they're a geometric way of looking at the Farey fractions (which are key in the circle method), so he said "well why don't I give it a shot"? Why does it work so magically in leading to quicker and cleaner integrals and approximation? What he's essentially doing is, instead of having each bit of the contour be an arc small enough that we integrate 'less' near the most prominent singularities, allowing the contour to approach the various singularities (and getting naturally closer to the ones with greater height - the 'less important' ones - faster). It's clear that this contour provides better bounds in proving the infinite sum form of $p(n)$ but in no way is it clear to me why, or how Rademacher first came up with the idea.
