Audio note: this article contains 162 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

One of the weirdest things in mathematics is how completely unrelated fields tend to connect with one another.
A particularly interesting case is the play between number theory (the study of natural numbers <span>_mathbb{N}={0,1,2dots}_</span>) and complex analysis (the study of functions on <span>_mathbb{C}={a+bi:a,bin mathbb{R}}_</span>). One is discrete and uses modular arithmetic and combinatorics; one is continuous and uses integrals and epsilon-delta proofs. And yet, many papers in modern number theory use complex analysis. Even if not directly, basically all of them rely on some result with a complex analytic flavor.

What's going on?

This post gives an example of a fundamental result in number theory that uses complex analysis, and tries to explain where that usage is coming from. My writing is semi-technical, I'm not trying to formalize everything, but mostly doing this as an exercise to clear out my own confusion (and hopefully to convince others of why this mathematical connection is valuable).