A gem of a demonstration breaks an 80-year-old record and offers new insights into prime numbers

The original version of this story appeared in Quanta Magazine.

Sometimes mathematicians try to tackle a problem head-on, and other times they try to tackle it sideways. This is especially true when the stakes in mathematics are high, as in the case of the Riemann hypothesis, the solution of which carries a million-dollar reward from the Clay Mathematics Institute. Its proof would give mathematicians much greater certainty about how prime numbers are distributed, while also entailing a host of other consequences, making it arguably the most important open question in mathematics.

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

in a innovative result Published online in May, Maynard and Larry Guth The Massachusetts Institute of Technology has set a new limit on the number of exceptions of a particular type, finally surpassing a record that had been 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 new test automatically leads to better approximations of how many primes exist in short intervals on the number line, and can offer many other insights into how prime numbers behave.

A careful side step

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

1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯.

This series will become arbitrarily large as more and more terms are added to it (mathematicians say it diverges). But if instead we were to add

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

you would get π²/6, or approximately 1.64. Riemann’s surprisingly powerful insight was to convert a series like this into a function, as follows:

ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + ⋯.

Then ζ(1) is infinite, but ζ(2) = π²/6.

Things get really interesting when you let s be a complex number, which has two parts: a “real” part, which is an everyday number, and an “imaginary” part, which is an everyday number multiplied by the square root of −1 (or Yo(as mathematicians write it). Complex numbers can be plotted on a plane, with the real part in the X-axis and the imaginary part on the and-axis. Here, for example, it is 3 + 4Yo.