A series of posts about Dirichlet’s Prime Number Theorem

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 \pi 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.

Structure of the series of posts

This series of posts will probably contain 4 posts:

  • We start by showing why the Riemann Zeta function diverges at s=1, i.e., why the harmonic series diverges
  • We then show why the Riemann Zeta function is equal to \pi^2/6 at s=2
  • Then we show why it diverges if we only sum up the reciprocal primes
  • Then, finally, we see why we cannot apply those methods for summing up primes in arithmetic progressions and present a proof of DPNT (which I might eventually split into 2 parts)

Part 1 – The Harmonic Series diverges

So let’s start by showing why the harmonic series diverges. As mentioned above, the harmonic series is what we get if we evaluate the Riemann Zeta function at s=1. It is \sum_{n=1}^\infty \frac{1}{n}.

We can write it down in the following way to see that it is in fact greater than an infinite sum of one-halfs: \underbrace{1}_{=2\frac{1}{2}} + \frac{1}{2} + \underbrace{\frac{1}{3}}_{>\frac{1}{4}} + \frac{1}{4} + \underbrace{\frac{1}{5}}_{>\frac{1}{8}} +  \underbrace{\frac{1}{6}}_{>\frac{1}{8}} +  \underbrace{\frac{1}{7}}_{>\frac{1}{8}} + \frac{1}{8} +  \ldots > \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \ldots. This sum of one-halfs goes on forever and therefore the harmonic series cannot converge because it is greater than that.

The fact that the harmonic series diverges tells us, by the way, that we can bridge arbitrary distances by stacking blocks like so:

Block stacking problem.svg

By cmglee, Anonimski – 16 wood samples.jpg, CC BY-SA 4.0, Link

In this picture, the center of gravity of a tower of any given size is always exactly aligned with the left edge of the block below it. This is nicely explained in this video:

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