The Mathematics of Origami

At the age of fifteen, in a copy of a book of logarithm tables he had been given as a present, Carl Friedrich Gauss wrote a small note. He had been counting prime numbers. In each interval of a thousand integers he wrote down how many primes he had found, and beside the count he wrote what he believed the count should have been. The guess was simple: out of the first x integers, about x / ln x are prime. He never proved it. He never even mentioned it in print. But the guess stood, more or less unchallenged, for almost a hundred years.

It is also wrong, in a very particular way. Gauss's estimate is too small, persistently and almost apologetically, by an amount that grows as you go further out. The error is not random. It carries within it the deepest unsolved problem in mathematics, and a piece of structure so unexpected that for a long time mathematicians did not know whether to take it seriously. The primes, which look at first like the most disorderly objects we can write down, turn out to be following something resembling a piece of music¹.

I — A first countWhat Gauss saw in a logarithm table

Write π(x) for the number of primes less than or equal to x. The function increases in unit jumps, one for each new prime. It is a staircase. Gauss's claim was that this staircase rises, on average, like the smooth curve x / ln x; better still, like the logarithmic integral Li(x), which is just the integral of 1/ln t from 2 up to x. The two curves are visually almost identical for the first many thousand integers, and Li(x) tracks π(x) more faithfully².

If you look closely, though, Li(x) sits a hair above π(x) at every plotted point. For a long time the evidence pointed in one direction: Gauss's curve overestimates the count, and so will any natural curve that smooths it. In 1914, Littlewood proved otherwise. Somewhere out beyond a number so large no one could plausibly compute it, π(x) is bigger than Li(x). And then it is smaller. And then bigger. The difference changes sign infinitely often. The smallest known place where it does so is around 10³⁰⁰. We have never seen the crossover. It is a theorem, not a sighting.

The difference between the primes and our best smooth guess changes sign infinitely often — but the smallest place we know of for sure is past every number we have ever counted to.

II — Riemann's leapA nine-page paper, and what it set loose

In 1859 Bernhard Riemann was elected to the Berlin Academy. The custom was to submit a short paper. Riemann sent in nine pages titled Über die Anzahl der Primzahlen unter einer gegebenen Größe — “On the number of primes less than a given quantity.” It is one of the strangest documents in the history of the subject. Its arguments are sketched, its computations partial, its conclusions stated with the casual confidence of a man writing to himself. It has set the agenda for analytic number theory ever since.

III — Hidden musicThe explicit formula

Why care about a curve's zeros? Because of a single, almost miraculous identity. From the zeros of ζ, Riemann showed how to write down a formula that recovers the prime count exactly. Not approximately, not on average: exactly. The formula has two pieces. The first is Gauss's smooth Li(x). The second is an infinite sum, one term for each nontrivial zero, in which each zero contributes a wave whose frequency is the height γ of the zero and whose amplitude decays slowly. Add them all up, and the staircase appears.

The primes are a staircase. The zeros of ζ are the tones that build the staircase. The hypothesis is that none of the tones is out of tune.

Marcus Chen wishes to thank the LMFDB collaboration.