riemann hypothesis 2022

A new approach to a $1 million mathematical enigma

Physicist translates the riemann zeta function into quantum field theory.

Numbers like π, e and φ often turn up in unexpected places in science and mathematics. Pascal's triangle and the Fibonacci sequence also seem inexplicably widespread in nature. Then there's the Riemann zeta function, a deceptively straightforward function that has perplexed mathematicians since the 19th century. The most famous quandary, the Riemann hypothesis, is perhaps the greatest unsolved question in mathematics, with the Clay Mathematics Institute offering a $1 million prize for a correct proof.

UC Santa Barbara physicist Grant Remmen believes he has a new approach for exploring the quirks of the zeta function. He has found an analogue that translates many of the function's important properties into quantum field theory. This means that researchers can now leverage the tools from this field of physics to investigate the enigmatic and oddly ubiquitous zeta function. His work could even lead to a proof of the Riemann hypothesis. Remmen lays out his approach in the journal Physical Review Letters.

"The Riemann zeta function is this famous and mysterious mathematical function that comes up in number theory all over the place," said Remmen, a postdoctoral scholar at UCSB's Kavli Institute for Theoretical Physics. "It's been studied for over 150 years."

An outside perspective

Remmen generally doesn't work on cracking the biggest questions in mathematics. He's usually preoccupied chipping away at the biggest questions in physics. As the fundamental physics fellow at UC Santa Barbara, he normally devotes his attention to topics like particle physics, quantum gravity, string theory and black holes. "In modern high-energy theory, the physics of the largest scales and smallest scales both hold the deepest mysteries," he remarked.

One of his specialties is quantum field theory, which he describes as a "triumph of 20 th century physics." Most people have heard of quantum mechanics (subatomic particles, uncertainty, etc.) and special relativity (time dilation, E=mc 2 , and so forth). "But with quantum field theory, physicists figured out how to combine special relativity and quantum mechanics into a description of how particles moving at or near the speed of light behave," he explained.

Quantum field theory is not exactly a single theory. It's more like a collection of tools that scientists can use to describe any set of particle interactions.

Remmen realized one of the concepts therein shares many characteristics with the Riemann zeta function. It's called a scattering amplitude, and it encodes the quantum mechanical probability that particles will interact with each other. He was intrigued.

Scattering amplitudes often work well with momenta that are complex numbers. These numbers consist of a real part and an imaginary part -- a multiple of √-1, which mathematicians call i . Scattering amplitudes have nice properties in the complex plane. For one, they're analytic (can be expressed as a series) around every point except a select set of poles, which all lie along a line.

"That seemed similar to what's going on with the Riemann zeta function's zeros, which all seem to lie on a line," said Remmen. "And so I thought about how to determine whether this apparent similarity was something real."

The scattering amplitude poles correspond to particle production, where a physical event happens that generates a particle with a momentum. The value of each pole corresponds with the mass of the particle that's created. So it was a matter of finding a function that behaves like a scattering amplitude and whose poles correspond to the non-trivial zeros of the zeta function.

With pen, paper and a computer to check his results, Remmen set to work devising a function that had all the relevant properties. "I had had the idea of connecting the Riemann zeta function to amplitudes in the back of my mind for a couple years," he said. "Once I set out to find such a function, it took me about a week to construct it, and fully exploring its properties and writing the paper took a couple months."

Deceptively simple

At its core, the zeta function generalizes the harmonic series:

This series blows up to infinity when x ≤ 1, but it converges to an actual number for every x > 1.

In 1859 Bernhard Riemann decided to consider what would happen when x is a complex number. The function, now bearing the name Riemann zeta, takes in one complex number and spits out another.

Riemann also decided to extend the zeta function to numbers where the real component was not greater than 1 by defining it in two parts: the familiar definition holds in places where the function behaves, and another, implicit definition covers the places where it would normally blow up to infinity.

Thanks to a theorem in complex analysis, mathematicians know there is only one formulation for this new area that smoothly preserves the properties of the original function. Unfortunately, no one has been able to represent it in a form with finitely many terms, which is part of the mystery surrounding this function.

Given the function's simplicity, it should have some nice features. "And yet, those properties end up being fiendishly complicated to understand," Remmen said. For example, take the inputs where the function equals zero. All the negative even numbers are mapped to zero, though this is apparent -- or "trivial" as mathematicians say -- when the zeta function is written in certain forms. What has perplexed mathematicians is that all of the other, non-trivial zeros appear to lie along a line: Each of them has a real component of ½.

Riemann hypothesized that this pattern holds for all of these non-trivial zeros, and the trend has been confirmed for the first few trillion of them. That said, there are conjectures that work for trillions of examples and then fail at extremely large numbers. So mathematicians can't be certain the hypothesis is true until it's proven.

But if it is true, the Riemann hypothesis has far-reaching implications. "For various reasons it crops up all over the place in fundamental questions in mathematics," Remmen said. Postulates in fields as distinct as computation theory, abstract algebra and number theory hinge on the hypothesis holding true. For instance, proving it would provide an accurate account of the distribution of prime numbers.

A physical analogue

The scattering amplitude that Remmen found describes two massless particles interacting by exchanging an infinite set of massive particles, one at a time. The function has a pole -- a point where it cannot be expressed as a series -- corresponding to the mass of each intermediate particle. Together, the infinite poles line up with the non-trivial zeros of the Riemann zeta function.

What Remmen constructed is the leading component of the interaction. There are infinitely more that each account for smaller and smaller aspects of the interaction, describing processes involving the exchange of multiple massive particles at once. These "loop-level amplitudes" would be the subject of future work.

The Riemann hypothesis posits that the zeta function's non-trivial zeros all have a real component of ½. Translating this into Remmen's model: All of the amplitude's poles are real numbers. This means that if someone can prove that his function describes a consistent quantum field theory -- namely, one where masses are real numbers, not imaginary -- then the Riemann hypothesis will be proven.

This formulation brings the Riemann hypothesis into yet another field of science and mathematics, one with powerful tools to offer mathematicians. "Not only is there this relation to the Riemann hypothesis, but there's a whole list of other attributes of the Riemann zeta function that correspond to something physical in the scattering amplitude," Remmen said. For instance, he has already discovered unintuitive mathematical identities related to the zeta function using methods from physics.

Remmen's work follows a tradition of researchers looking to physics to shed light on mathematical quandaries. For instance, physicist Gabriele Veneziano asked a similar question in 1968: whether the Euler beta function could be interpreted as a scattering amplitude. "Indeed it can," Remmen remarked, "and the amplitude that Veneziano constructed was one of the first string theory amplitudes."

Remmen hopes to leverage this amplitude to learn more about the zeta function. "The fact that there are all these analogues means that there's something going on here," he said.

And the approach sets up a path to possibly proving the centuries-old hypothesis. "The innovations necessary to prove that this amplitude does come from a legitimate quantum field theory would, automatically, give you the tools that you need to fully understand the zeta function," Remmen said. "And it would probably give you more as well."

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• 11 November 2022
• Correction 14 November 2022

Mathematician who solved prime-number riddle claims new breakthrough

• Davide Castelvecchi

You can also search for this author in PubMed   Google Scholar

A mathematician who went from obscurity to luminary status in 2013 for cracking a century-old question about prime numbers now claims to have solved another. The problem is similar to — but distinct from — the Riemann hypothesis, which is considered one of the most important problems in mathematics.

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Nature 611 , 645-646 (2022)

doi: https://doi.org/10.1038/d41586-022-03689-2

Updates & Corrections

Correction 14 November 2022 : This article has been amended to clarify that the formula Zhang claims to have proved is not the Landau-Siegel zeros conjecture, but a weaker version of it.

Zhang, Y. Preprint at https://arxiv.org/abs/2211.02515 (2022).

Zhang, Y. Preprint at https://arxiv.org/abs/0705.4306 (2007).

Zhang, Y. Ann. Math. 179 , 1121–1174 (2014).

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Mathematicians report possible progress on proving the riemann hypothesis.

A new study of Jensen polynomials revives an old approach

STILL ELUSIVE   Researchers may have edged closer to a proof of the Riemann hypothesis — a statement about the Riemann zeta function, plotted here — which could help mathematicians understand the quirks of prime numbers.

Jan Homann/Wikimedia Commons

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By Emily Conover

May 24, 2019 at 12:03 pm

Researchers have made what might be new headway toward a proof of the Riemann hypothesis, one of the most impenetrable problems in mathematics. The hypothesis, proposed 160 years ago, could help unravel the mysteries of prime numbers.

Mathematicians made the advance by tackling a related question about a group of expressions known as Jensen polynomials, they report May 21 in Proceedings of the National Academy of Sciences . But the conjecture is so difficult to verify that even this progress is not necessarily a sign that a solution is near ( SN Online: 9/25/18 ).

At the heart of the Riemann hypothesis is an enigmatic mathematical entity known as the Riemann zeta function. It’s intimately connected to prime numbers — whole numbers that can’t be formed by multiplying two smaller numbers — and how they are distributed along the number line. The Riemann hypothesis suggests that the function’s value equals zero only at points that fall on a single line when the function is graphed, with the exception of certain obvious points. But, as the function has infinitely many of these “zeros,” this is not easy to confirm. The puzzle is considered so important and so difficult that there is a $1 million prize for a solution , offered up by the Clay Mathematics Institute.

But Jensen polynomials might be a key to unlocking the Riemann hypothesis. Mathematicians have previously shown that the Riemann hypothesis is true if all the Jensen polynomials associated with the Riemann zeta function have only zeros that are real, meaning the values for which the polynomial equals zero are not imaginary numbers — they don’t involve the square root of negative 1. But there are infinitely many of these Jensen polynomials.

Studying Jensen polynomials is one of a variety of strategies for attacking the Riemann hypothesis. The idea is more than 90 years old, and previous studies have proved that a small subset of the Jensen polynomials have real roots. But progress was slow, and efforts had stalled.

Now, mathematician Ken Ono and colleagues have shown that many of these polynomials indeed have real roots, satisfying a large chunk of what’s needed to prove the Riemann hypothesis.

“Any progress in any direction related to the Riemann hypothesis is fascinating,” says mathematician Dimitar Dimitrov of the State University of São Paulo. Dimitrov thought “it would be impossible that anyone will make any progress in this direction,” he says, “but they did.”

It’s hard to say whether this progress could eventually lead to a proof. “I am very reluctant to predict anything,” says mathematician George Andrews of Penn State, who was not involved with the study. Many strides have been made on the Riemann hypothesis in the past, but each advance has fallen short. However, with other major mathematical problems that were solved in recent decades, such as Fermat’s last theorem ( SN: 11/5/94, p. 295 ), it wasn’t clear that the solution was imminent until it was in hand. “You never know when something is going to break.”

The result supports the prevailing viewpoint among mathematicians that the Riemann hypothesis is correct. “We’ve made a lot of progress that offers new evidence that the Riemann hypothesis should be true,” says Ono, of Emory University in Atlanta.

If the Riemann hypothesis is ultimately proved correct, it would not only illuminate the prime numbers, but would also immediately confirm many mathematical ideas that have been shown to be correct assuming the Riemann hypothesis is true.

In addition to its Riemann hypothesis implications, the new result also unveils some details of what’s known as the partition function , which counts the number of possible ways to create a number from the sum of positive whole numbers ( SN: 6/17/00, p. 396 ). For example, the number 4 can be made in five different ways: 3+1, 2+2, 2+1+1, 1+1+1+1, or just the number 4 itself.

The result confirms an earlier proposition about the details of how that partition function grows with larger numbers. “That was an open question … for a long time,” Andrews says. The real prize would be proving the Riemann hypothesis, he notes. That will have to wait.

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The Riemann Hypothesis: worth a billion dollars?

As anyone who mentions the Riemann Hypothesis in a research grant knows, the Clay Mathematics Institute offers a million-dollar prize for a proof. I shall outline some of the many roadblocks to a proof and hope to convince the audience that the prize offered is off by three orders of magnitude.

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Home — News — 2024 Clay Research Award

2024 Clay Research Award

Date: 01 May 2024

James Newton and Jack Thorne

A Clay Research Award is made to James Newton (Oxford) and Jack Thorne (Cambridge) in recognition of their remarkable proof of the existence of the symmetric power functorial lift for Hilbert modular forms.

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Paul Nelson

A Clay Research Award is made to Paul Nelson (Aarhus) in recognition of his groundbreaking contributions to the analytic theory of automorphic forms.  His work has resulted in the first convexity-breaking bounds for a large class of L-functions on the critical line (including all the standard ones of GL(n)). This marks a signficant advance in a field initiated one hundred years ago by Hermann Weyl in the context of the Riemann Zeta function.  Nelson analyzes L-values via certain associated automorphic periods. His powerful approach involves many ingredients, including a refinement of the orbit method that he developed in earlier work with Venkatesh, and an analysis of the geometric side of an appropriate Relative Trace Formula, which facilitates the use of Amplification.

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The Clay Mathematics Institute is pleased to announce that Ishan Levy and Mehtaab Sawhney have been awarded Clay Research Fellowships. Ishan Levy will receive his PhD from the Massachusetts Institute of Technology in 2024, under the supervision of Michael Hopkins. Ishan has been appointed as a Clay Research Fellow for five years beginning 1 July 2024. Mehtaab […]

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