What the Riemann hypothesis is, and why it matters so much

Ihave watched a great many videos about the Riemann hypothesis. Most either bury you in definitions from the first minute or never touch the mathematics at all and just keep repeating that primes are mysterious. This video, which Quanta Magazine commissioned from the Rutgers mathematician Alex Kontorovich, does neither, and to my mind it is the single best explanation available right now. In sixteen minutes it takes someone with no advanced mathematics and leaves them genuinely understanding the greatest unsolved problem in the field. So what is the thing that deserves all this trouble?

The Riemann hypothesis is one of the seven Millennium Problems of the Clay Mathematics Institute, and whoever proves it not only enters the history of mathematics but collects a million-dollar prize. The money is not really the point. This hypothesis answers a question as old as mathematics itself. How are the prime numbers distributed along the number line? Primes are the building blocks of every number, yet they appear with no visible order, and there is no formula that tells you where the next one will show up.

The story begins before Riemann. In the late 1700s a young Gauss computed by hand every prime up to three million, filled his tables with them, and sensed that the thinning out of the primes obeys a rule, the ratio 1/log x. Around the same time Euler discovered that the zeta function can also be written as an infinite product with one factor for every prime, and he was the first to see a hidden bond between that function and the primes. The man who joined the two clues was Bernhard Riemann. In the eight-page paper he published in 1859 he carried the zeta function into the complex plane and proved that the points where the function takes the value zero control the distribution of the primes down to the letter. Then he wrote the famous sentence. All the important zeros lie on a single vertical line in the complex plane, the critical line, where the real part is exactly one half. That is the whole hypothesis. Riemann died without proving it, and from that day to this nobody else has managed to either.

Not for want of trying. For more than 150 years the biggest names in mathematics have wrestled with this problem, and computers have been thrown at it as well. For a time an enormous distributed computing project was checking more than a billion zeros a day, and what it was hunting for was a single fugitive, because one zero falling off the line would demolish the hypothesis on the spot. Every one of the ten trillion zeros checked so far has landed on the critical line. But a pattern that holds ten trillion times is no guarantee that it holds forever, and no machine can count forever. Brute force will never finish this job. One road remains, and it is the same road the ancient Greeks walked. A rigorous mathematical proof.

And if that proof ever arrives, what happens? Countless theorems, in fields as far apart as cryptography and quantum physics, have been proved on the assumption that the Riemann hypothesis is true, which means an entire edifice is resting on a single unproven sentence. A proof would turn all those conditional results into settled theorems overnight, and we would know everything we have any right to know about how the primes are distributed. That follows from what Riemann showed. Each zeta zero adds one harmonic to the prime counting function, and when you sum them all you get a formula that fits the jagged staircase of the primes perfectly. Pinning down the zeros means pinning down the primes.

Kontorovich tells this whole story with remarkable patience. He has you picture the primes as a city of single-storey houses, turns Gauss’s staircase into a smooth curve before your eyes, builds up complex numbers and analytic continuation from nothing, and in the finale shows how the zeta zeros construct that staircase one harmonic at a time. The spiral at the top of this piece is the video’s opening image. Set aside sixteen minutes this weekend. It is worth it.