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 _mathbb{N}={0,1,2dots}_) and complex analysis (the study of functions on _mathbb{C}={a+bi:a,bin mathbb{R}}_). 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).

Technical details in the proof follow Ang Li's notes; exposition, intuition and possible mistakes [...]

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Outline:

(01:31 ) I Dirichlets Theorem

(03:58 ) II Eulers proof for infinitude of primes

(08:12 ) III Trying the same for Dirichlet

(16:25 ) IV Wrapping up (or: getting started)

(19:20 ) V Looking ahead

The original text contained 4 footnotes which were omitted from this narration.

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First published:

December 20th, 2025

Source:

https://www.lesswrong.com/posts/hRJ72iSzeACkPGvHt/the-unreasonable-deepness-of-number-theory

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Narrated by TYPE III AUDIO.

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