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When numbers are plotted in a polar pattern called a Sack’s spiral, the distribution of primes (black dots) hints at some hidden structure.Claudio Rocchini/Wikimedia Commons CC-BY

A version of this story appeared in Science, Vol 385, Issue 6708.Download PDF

Just as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors—those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.

"At first sight, they look pretty random," says James Maynard, a mathematician at the University of Oxford. "But actually, there’s believed to be this hidden structure within the prime numbers."

For 165 years, mathematicians seeking that structure have focused on the Riemann hypothesis. Proving it would offer a Rosetta Stone for decoding the primes—as well as a $1 million award from the Clay Mathematics Institute. Now, in a preprint posted online on 31 May, Maynard and Larry Guth of the Massachusetts Institute of Technology have taken a step in this direction by ruling out certain exceptions to the Riemann hypothesis. The result is unlikely to win the cash prize, but it represents the first progress in decades on a major knot in math’s biggest unsolved problem, and it promises to spark new advances throughout number theory.

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"It’s a sensational breakthrough," says Alex Kontorovich, a mathematician at Rutgers University. "There are a bunch of new ideas going into this proof that people are going to be mining for years."

Predicting exactly where the next prime will show up on the number line is challenging, but describing the cumulative abundance of primes over large intervals is surprisingly straightforward. In the late 1700s, at the age of 16, German mathematician Carl Friedrich Gauss saw that the frequency of prime numbers seems to diminish as they get bigger and posited that they scale according to a simple formula: the number of primes less than or equal to X is roughly X divided by the natural logarithm of X. Gauss’s estimate has stood up impressively well. To the best mathematicians can tell, the actual number of primes bounces slightly above and below this curve up to infinity. That known primes follow such a simple formula so closely suggests the primes aren’t completely random; there must be some deep connections governing where they appear.

But mathematicians want to know exactly how well Gauss’s guess holds up—and why. In 1859, Bernhard Riemann, another renowned German mathematician, sought help from a different function, now called the Riemann zeta function. For inputs, the function takes complex numbers, which are a combination of real numbers and what mathematicians call "imaginary" ones: a normal number multiplied by the square root of –1. The function seems to capture the discrepancies between Gauss’s curve and the real distribution of primes. The places where Riemann’s function equals zero—referred to as zeta zeros—directly describe the fluctuating errors around Gauss’s curve.

Here, Riemann made his famous conjecture: ignoring certain trivial solutions for negative inputs, all the zeta zeros should exist for inputs where the real part is one-half. If his hypothesis is true, it means the seemingly random fluctuations in the abundance of primes are bounded, leaving no big clumps or gaps in their distribution along the number line. Any proof of the Riemann hypothesis would be a window into the secret clockwork governing the primes’ irregular pattern. It would offer a chance to "reverse-engineer the random number generator of the primes," says Maksym Radziwill, a mathematician at Northwestern University.

To date, mathematicians have used computers to test more than 10 trillion nontrivial zeta zeros—and they all lie at exactly one-half. But no amount of empirical evidence will satisfy mathematicians: They want a formal proof the zeros can never lie anywhere else. Although no one suspects the Riemann hypothesis to be false, "A proof gives much more than just a statement being true," Maynard says. "It gives an understanding as to why it’s true, so you have some powerful new technique for understanding prime numbers."

After 165 years, mathematicians remain "completely stumped" as to how they might prove the Riemann hypothesis, Maynard says. "We don’t even have a plausible line of attack." So, they’ve resorted to taking smaller bites out of the problem by determining where zeta zeros can’t be.

Mathematicians already know that the nontrivial zeta zeros are confined between 0 and 1. They also know about a mirror symmetry around one-half, whereby ruling out zeta zeros at three-quarters would also rule them out at one-quarter. So some techniques focused on the region from one-half to three-quarters whereas others worked better between three-quarters and 1. This left a small but unsettling possibility that many zeros could be hiding out right at three-quarters.

The best bound for how many zeros can lie at three-quarters came from the British mathematician Albert Ingham in 1940. No one has done better since. "It was a bit outrageous that this [limit] could not be lowered," Radziwill says. "Basically, nobody was working on this because everybody gave up."

Save for Maynard, a 37-year-old virtuoso who specializes in analytic number theory, for which he won the 2022 Fields Medal—math’s most prestigious award. In dedicated Friday afternoon thinking sessions, he returned to the problem again and again over the past decade, to no avail. At an American Mathematical Society meeting in 2020, he enlisted the help of Guth, who specializes in a technique known as harmonic analysis, which draws from ideas in physics for separating sounds into their constituent notes. Guth also sat with the problem for a few years. Just before giving up, he and Maynard hit a break. Borrowing tactics from their respective mathematical dialects and exchanging ideas late into the night over an email chain, they pulled some unorthodox moves to finally break Ingham’s bound.

Radziwill says the work represents the first new idea in the hunt for zeta zeros in 50 years. "This might actually restart an area that was really neglected for a long time," he says. "I mean, there could be a Renaissance."

The improved bound does little to help mathematicians prove the Riemann hypothesis overall. But Radziwill and Kontorovich expect the result will ripple throughout number theory. The new constraint immediately allows mathematicians to better estimate the number of primes in shorter intervals, for instance.

But the real impact lies in the maneuvers that allowed Guth and Maynard to break the barrier, fresh tools that may well apply beyond prime number theory, Radziwill says. He suggests the new strategies may help simplify some of his prior work on dynamical systems, and they could also help with another longstanding conjecture known as the Kakeya problem, in which a shifting needle is spun through 360°, tracing out complicated circular or deltoid shapes while covering the least possible area. Guth, meanwhile, is most excited about using these ideas to explore the deep relationship between the physics of waves and the distribution of number sets.

Looking back, Guth recalls a quote from the Austrian poet Rainer Maria Rilke, who instructs an aspiring poet to "live the questions" rather than seeking answers. For Guth, this strategy of being comfortably uncomfortable with intractable problems resonates with his experience as a mathematician.

"I don’t at all expect to resolve the Riemann hypothesis," he says. "But we hope that wondering about something we don’t understand will help find something that is beautiful or maybe even useful."

Editor’s note, 31 July, 1:40 p.m.: The picture accompanying this story was changed to better accommodate readers with color blindness.

doi: 10.1126/science.z0p8ow7

Zack Savitsky is a science journalist specializing in the physical sciences.

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