‘Gem’ of Proof Breaks 80-Year-Ancient Record, Offers Novel Look at Primes

Original version With this history appeared in Quanta Magazine.

Sometimes mathematicians try to approach a problem head-on, and sometimes they approach it sideways. This is especially true when the mathematical stakes are high, as in the case of the Riemann hypothesis, which carries a $1 million prize from the Clay Mathematics Institute. Its proof would give mathematicians much greater certainty about how the primes are distributed, while also implying a host of other consequences—making it perhaps the most vital open question in mathematics.

Mathematicians have no idea how to prove the Riemann hypothesis. But they can still get useful results by simply showing that the number of possible exceptions to it is confined. “In many cases, this may be as good as the Riemann hypothesis itself,” he said. James Maynard from the University of Oxford. “We can get similar results about primes from this.”

IN a breakthrough result published online in May, Maynard and Larry Guth The Massachusetts Institute of Technology has set a modern limit on the number of exceptions of a certain type, finally breaking a record set more than 80 years earlier. “It’s a sensational result,” he said Henryk Iwaniec from Rutgers University. “It’s very, very, very difficult. But it’s a gem.”

The modern proof automatically leads to better approximations for the number of primes that appear at brief intervals on the number line, and may also provide many other insights into the behavior of primes.

Careful avoidance

The Riemann hypothesis is a statement about the central pattern in number theory called the Riemann zeta function. The zeta function (ζ) is a generalization of the basic sum:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯.

1 + 1/2² + 1/3² + 1/4² +1/5² + ⋯ = 1 + 1/4 + 1/9 + 1/16 + 1/25 +⋯

(G)S) = 1 + 1/2^S + 1/3^S + 1/4^S +1/5^S + ⋯.

So ζ(1) is infinite, but ζ(2) = π²/6.

Things get really fascinating when you let them. S let be a complicated number consisting of two parts: a “real” part, which is the everyday number, and an “imaginary” part, which is the everyday number multiplied by the square root of −1 (or AND(as mathematicians write). Convoluted numbers can be represented on a plane, with the real part on X-axis and imaginary part on and-os. Here for example it is 3 + 4AND.