Mathematicians Are Edging Close to Solving One of the World's 7

In new research, mathematicians have narrowed down one of the biggest outstanding problems in math.
Huge breakthroughs in math and science are usually the work of many people over many years.
Seven math problems were given a $1 million bounty each in 2000, and just one has been solved so far.

A new preprint math paper is lighting up the airwaves as mathematicians tune in for a possible breakthrough in a very old, very sticky problem in number theory. Riemann’s hypothesis—concerning the distribution of prime numbers throughout the number line—dates back over 160 years. While the new paper doesn’t purport to solve the problem, it could be a substantial step toward a solution. That could enable other number theorists to keep taking steps toward solving it, and (perhaps more importantly) winning a $1 million prize.

The “Millennium Problems” are seven infamously intractable math problems laid out in the year 2000 by the prestigious Clay Institute, each with $1 million attached as payment for a solution. They span all areas of math, as the Clay Institute was founded in 1998 to push the entire field forward with financial support for researchers and important breakthroughs.

But the only solved Millennium Problem so far, the Poincare conjecture, illustrates one of the funny pitfalls inherent to offering a large cash prize for math. The winner, Grigori Perelman, refused the Clay prize as well as the prestigious Fields Medal. He withdrew from mathematics and public life in 2006, and even in 2010, he still insisted his contribution was the same as the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.

Math, all sciences, and arguably all human inquiries are filled with pairs or groups that circle the same finding at the same time until one officially makes the breakthrough. Think about Sir Isaac Newton and Gottfried Leibniz, whose back-and-forth about calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow discoverers of DNA, James Watson and Francis Crick. Even the Bechdel Test for women in media is sometimes called the Bechdel-Wallace Test, because humans are almost always in collaboration.

That’s what makes this new paper so important. Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are formed and how they reach out into the number line. Maynard is just 37, and won the Fields Medal himself in 2022. Guth, a decade older, has won a number of important prizes with a little less name recognition.

The Riemann hypothesis is not directly related to prime numbers, but it has implications that ripple through number theory in different ways (including with prime numbers). Basically, it deals with where and how the graph of a certain function of complex numbers crosses back and forth across axes. The points where the function crosses an axis is called a “zero,” and the frequency with which those zeroes appear is called the zero density.

In the far reaches of the number line, prime numbers become less and less predictable (in the proverbial sense). They are not, so far, predictable in the literal sense—a fact that is an underpinning of modern encryption, where data is protected by enormous strings of integers made by multiplying enormous prime numbers together. The idea of a periodic table of primes, of any kind of template that could help mathematicians better understand where and how large primes cluster together or not, is a holy grail.

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they’ve proven that these polynomials have a certain number of large values, or solutions, within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven’t found the perfect prescription. “If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,” Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value—as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

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