The Riemann--Hilbert correspondence, roughly, tells you that you can convert differential equations into certain geometric objects, and that this conversion process loses no information. In particular, one can convert questions about differential equations into geometric questions, and conversely one can convert certain geometric questions into problems about differential equations.

This article gives a simple example of the Riemann--Hilbert correspondence.

There seems to be a lot of online posts/videos which describe the zeta function (and how you can earn 1 million dollars for understanding something about its zeroes). But these posts often don't explain what the zeta function actually has to do with the distribution of the prime numbers.

My friend and I tried to write an explanation, using only high school level mathematics, of how you can understand the prime numbers using the zeta function.