‘Sensational’ Proof Delivers New Insights Into Prime Numbers

Sometimes mathematicians try to tackle a problem head on, and sometimes they come at it sideways. That’s especially true when the mathematical stakes are high, as with the Riemann hypothesis, whose solution comes with a $1 million reward from the Clay Mathematics Institute. Its proof would give mathematicians much deeper certainty about how prime numbers are distributed, while also implying a host of other consequences — making it arguably the most important open question in math.

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

In a breakthrough result posted online in May, Maynard and Larry Guth of the Massachusetts Institute of Technology established a new cap on the number of exceptions of a particular type, finally beating a record that had been set more than 80 years earlier. “It’s a sensational result,” said Henryk Iwaniec of Rutgers University. “It’s very, very, very hard. But it’s a gem.”

The new proof automatically leads to better approximations of how many primes exist in short intervals on the number line, and stands to offer many other insights into how primes behave.

A Careful Sidestep

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

$latex 1 + frac{1}{2} + frac{1}{3} + frac{1}{4} + frac{1}{5} + cdots $.

This series will become arbitrarily large as more and more terms are added to it — mathematicians say that it diverges. But if instead you were to sum up

$latex 1 + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + frac{1}{5^2} + cdots=1 + frac{1}{4} + frac{1}{9} + frac{1}{16} + frac{1}{25} + cdots $

you would get $latex frac{pi^2}{6}$, or about 1.64. Riemann’s surprisingly powerful idea was to turn a series like this into a function, like so:

$latex zeta (s)=1 + frac{1}{2^s} + frac{1}{3^s} + frac{1}{4^s} + frac{1}{5^s} + cdots$.

So $latex zeta (1)$ is infinite, but $latex zeta (2)=frac{pi^2}{6}$.

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 i, as mathematicians write it). Complex numbers can be plotted on a plane, with the real part on the x-axis and the imaginary part on the y-axis. Here, for example, is 3 + 4i.

The zeta function takes points on the complex plane as inputs, and it produces other complex numbers as outputs. It turns out that for some complex numbers, the zeta function is equal to zero. Figuring out where those zeros are located on the complex plane is one of the most interesting questions in mathematics.

In 1859, Bernhard Riemann conjectured that all the zeros are concentrated on two lines. If you extend the zeta function so you can compute it for negative inputs, you’ll find that it equals zero for all negative even numbers: −2, −4, −6 and so on. This is relatively easy to show, so these are called trivial zeros. Riemann conjectured that all the other zeros of the function, called nontrivial zeros, have a real part of 1/2, and so are located on this vertical line.

This is the Riemann hypothesis, and proving it has been prohibitively difficult. Mathematicians know that every nontrivial zero must have a real part between zero and 1, but they can’t rule out that some zeros might have a real part of, say, 0.499.

What they can do is show that such zeros must be incredibly rare. In 1940, an English mathematician named Albert Ingham established an upper bound on the number of zeros whose real part is not equal to 1/2 that mathematicians continue to use as a point of reference today.

A few decades later, in the 1960s and ’70s, other mathematicians figured out how to translate Ingham’s result into statements about how clumped or spread out prime numbers are as you move further along the number line, and about other patterns they might form. Around the same time, mathematicians also introduced new techniques that improved Ingham’s bounds for zeros with a real part greater than 3/4.

But it turned out that the most important zeros to cap were those with a real part of exactly 3/4. “Lots of headline results about prime numbers were limited by our understanding of zeros with real part 3/4,” Maynard said.

About a decade ago, Maynard started thinking about how to improve Ingham’s estimate for those particular zeros. “It’s been one of my favorite problems in analytic number theory,” he said. “It always felt tempting that you just have to work a bit harder, and you’ll be able to get an improvement.” But year after year, no matter how many times he came back to it, he kept getting stuck. “It almost sucked you in, and it looked much more innocent than I think it was.”

Then, in early 2020, during a plane trip to a conference in Colorado, an idea came to him. Perhaps, Maynard thought, tools from another area of math called harmonic analysis might be useful.

Larry Guth, an expert in harmonic analysis who was at the same conference, just happened to already be thinking along similar lines. “But I didn’t know the analytic number theory at all well,” he said. Maynard explained the number theory side of the story to him over lunch and gave him a test case to work with. Guth studied it on and off for a few years, only to realize that his techniques from harmonic analysis wouldn’t work.

But he didn’t stop thinking about the problem, and he experimented with new approaches. He got back in touch with Maynard in February. The two started collaborating in earnest, combining their different perspectives. A few months later, they had their result.

A Mathematical Gambit

Guth and Maynard started out by converting the problem they wanted to solve into another one. If you have a zero that doesn’t have a real part of 1/2, then a related function, called a Dirichlet polynomial, must produce a very large output. As a result, proving that there are few exceptions to the Riemann hypothesis is equivalent to showing that the Dirichlet polynomial cannot get large too often.

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