# The Prime Bet

Let's say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet:

We'll choose two positive integers at random. If they have any divisor in common (other than $1$) I'll pay you a dollar, else you'll pay me a dollar. Are you in?

Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you?

When two integers have no divisors in common except the trivial divisor $1$ we say they are coprime or relatively prime. $6$ and $9$ have the common divisor $3$, so they are not coprime, whilst $8$ and $9$ only have the trivial common divisor $1$, so they are coprime.

This makes you start thinking: "As numbers grow bigger, aren't there a lot of divisors out there? After all, half the numbers are even, so if we hit two even numbers, they'll have the factor $2$ in common and I'll win. And then there's $3$, $5$, $7$, ... Seems like a good deal!" Continue reading The Prime Bet

# Visualising the Riemann Hypothesis

One stubborn source of frustration when working with complex numbers is the fact that visualisation becomes tedious, if not impossible. Complex numbers have 2 "real" dimensions in themselves, which give rise to the complex plane. That's all good and fair. But if you talk about a function with complex domain and codomain, you already deal with a 4-dimensional graph. Unfortunately, my mind can only handle 3 dimensions (on a good day). One can resort to taking the absolute value of the function instead, or map real and imaginary part individually, resulting in a 3-dimensional graph, but all of these solutions fail to satisfy in one respect or another.

However, there is one more dimension we can exploit: time! Used in the right way, this can produce wonderful videos like this one:

# Applying the Explicit Formula

It's quite some time since we arrived at Riemann's main result, the explicit formula

where $J(x)$ is the prime power counting function introduced even earlier. It's high time we applied this!

First, let's take a look at $J(x)$ when calculating it exactly: