Inspired by a video by 3blue1brown, I created a series of posts that leads up to a proof of Dirichlet’s Prime Number Theorem. Here’s the original video that inspired me to do this:
The video visualizes tuples of two identical numbers, both primes, and interprets them as polar coordinates (distance from the origin and angle off the x-axis). So, for example, the point (3, 3) lies almost on the negative x-axis (because an angle of corresponds to a half-turn) and has 3 units distance from the origin. If you plot many (hundreds, thousands) of integers or only prime numbers that way, then you can see different patterns at different scales.
The video then discusses how these patterns, that are visible at different scales, arise and finishes by mentioning Dirichlet’s Prime Number Theorem (DPNT) which says that every arithmetic progression contains infinitely many prime numbers.
To me, that felt like a good motivation to understand some more math. I consider myself a hobby mathematician and I often use videos by 3blue1brown or Numberphile as starting points to explore some more math. That’s what motivated me to explore this topic.(more…)