If there is a God, this is the closest you’ll ever come to hearing Him sing. Let me explain.
The greatest mystery in arithmetic is: How are the prime numbers distributed?
We can start by drawing a “prime staircase” that jumps up at each of the prime numbers, beginning with 2, 3, 5, 7, and 11:
Here’s how that staircase looks when you zoom out far enough to see all the primes up to 100:
Understanding the distribution of the primes means understanding the shape of this staircase.
For that, we try superimposing a nice smooth curve that tracks the staircase tolerably well:
Now the problem is to understand the discrepancies—the difference between the staircase itself and the smooth red curve. Some of those discrepancies are positive (when the staircase is above the curve) and others are negative. If you graph the discrepancies, they look quite chaotic. Here’s what they look like, for example, in the range from 1 million to 2 million:
The great 19th century mathematician Bernhard Riemann suggested that we can resolve the chaos by decomposing it into a sum of simple waves.
To see how that could work, here are three nice simple waves:
If you add these three together, you get something that’s already starting to look a little chaotic:
Riemann’s hypothesis was that if you add up infinitely many waves instead of just 3, you should be able to reproduce the extreme chaos of the prime staircase discrepancies. If that’s true, the apparent chaos is not so chaotic after all.
(I should mention that everything I’ve said here is true in spirit but false in technical detail. I am sacrificing accuracy for clarity.)
Now a wave can always be interpreted as a musical note—the distance from peak to peak represents a tone and the height of the peaks represents the volume. So the Riemann hypothesis says in essence that the distribution of primes—the very foundation of arithmetic—is encoded in a piece of music.
We do not know that Riemann’s hypothesis is true—and in fact, proving (or disproving) it is widely considered the single most important unsolved problem in all of mathematics—but we have vast evidence for it, and assuming it is true, we know the first several billion notes.
How do those notes sound? You can listen to them in this MP3 file (the same one I linked to in the first line of this post), which I took from the web page of the mathematician Jeffrey Stopple. (For readers who care about precision, Stopple’s pages are an excellent antidote to the cavalier attitude I’ve taken in this post.) The notes are added one at a time, at intervals of .2 seconds. At the end, all 100 notes play together for 10 seconds. That final 10-second tone represents the sum of the first 100 Riemann waves, which in turn gives a pretty good approximation to the “chaos” of the discrepancies from the prime staircase. This is the music of the primes.