Tag Archives: Zeta function

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:

Continue reading Visualising the Riemann Hypothesis

Applying the Explicit Formula

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

J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,

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:

J(x) Continue reading Applying the Explicit Formula