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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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Riemann Hypothesis

A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix).

Proof of the Riemann hypothesis is number 8 of Hilbert's problems and number 1 of Smale's problems .

In 2000, the Clay Mathematics Institute ( http://www.claymath.org/ ) offered a $1 million prize ( http://www.claymath.org/millennium/Rules_etc/ ) for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line ), does not earn the $1 million award.

The Riemann hypothesis is equivalent to the statement that all the zeros of the Dirichlet eta function (a.k.a. the alternating zeta function)

By modifying a criterion of Robin (1984), Lagarias (2000) showed that the Riemann hypothesis is equivalent to the statement that

There is also a finite analog of the Riemann hypothesis concerning the location of zeros for function fields defined by equations such as

According to Fields medalist Enrico Bombieri, "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers" (Havil 2003, p. 205).

In Ron Howard's 2001 film A Beautiful Mind , John Nash (played by Russell Crowe) is hindered in his attempts to solve the Riemann hypothesis by the medication he is taking to treat his schizophrenia.

In the Season 1 episode " Prime Suspect " (2005) of the television crime drama NUMB3RS , math genius Charlie Eppes realizes that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security.

In the novel Life After Genius (Jacoby 2008), the main character Theodore "Mead" Fegley (who is only 18 and a college senior) tries to prove the Riemann Hypothesis for his senior year research project. He also uses a Cray Supercomputer to calculate several billion zeroes of the Riemann zeta function. In several dream sequences within the book, Mead has conversations with Bernhard Riemann about the problem and mathematics in general.

Portions of this entry contributed by Len Goodman

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Has one of math’s greatest mysteries, the Riemann hypothesis, finally been solved?

Professor of Mathematics, University of Richmond

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Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis .

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002 .

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

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2.5: The Riemann Hypothesis

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Definition 2.19

The Riemann zeta function \(\zeta(z)\) is a complex function defined as follows on \(\{z \in \mathbb{C} | \mbox{Re}z > 1\}\)

\[\zeta (z) = \sum_{n=1}^{\infty} n^{-z} \nonumber\]

On other values of \(z \in \mathbb{C}\) it is defined by the analytic continuation of this function (except at \(z = 1\) where it has a simple pole).

Analytic continuation is akin to replacing \(e^x\) where \(x\) is real by \(e^z\) where \(z\) is complex. Another example is the series \(\sum_{j=0}^{\infty} z^j\). This series diverges for \(|z| > 1\). But as an analytic function, it can be replaced by \((1-z)^{-1}\) on all of \(\mathbb{C}\) except at the pole \(z = 1\) where it diverges.

Recall that an analytic function is a function that is differentiable. Equivalently, it is a function that is locally given by a convergent power series. If \(f\) and \(g\) are two analytic continuations to a region \(U\) of a function \(h\) given on a region \(V \subset U\), then the difference \(f-g\) is zero on some \(U\) and therefore all its power expansions are zero and so it must be zero on the the entire region. Hence, analytic conjugations are unique. That is the reason they are meaningful. For more details, see for example [ 4 , 14 ].

It is customary to denote the argument of the zeta function by \(s\). We will do so from here on out. Note that \(|n-s| = n-\mbox{Re} s\), and so for \(\mbox{Re} s > 1\) the series is absolutely convergent. At this point, the student should remember – or look up in [ 23 ] – the fact that absolutely convergent series can be rearranged arbitrarily without changing the sum. This leads to the following proposition.

Proposition 2.20

For \(\mbox{Re} s > 1\) we have

\[\sum_{n=1}^{\infty} n-s = \prod_{p prime} (1-p^{-s})^{-1} \nonumber\]

There are two common proofs of this formula. It is worth presenting both.

The first proof uses the Fundamental Theorem of Arithmetic. First, we recall that use geometric series

\[(1-p^{-s})^{-1} = \sum_{k=0}^{\infty} p^{-ks} \nonumber\]

to rewrite the right hand of the Euler product. This gives

\[\prod_{p prime} (1-p^{-s})^{-1} = (\sum_{k_1 = 0}^{\infty} p_{1}^{-k_{1}s}) (\sum_{k_2 = 0}^{\infty} p_{2}^{-k_{2}s}) (\sum_{k_3 = 0}^{\infty} p_{3}^{-k_{3}s}) \nonumber\]

Re-arranging terms yields

\[\dots = (p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}} \dots)^{-s} \nonumber\]

By the Fundamental Theorem of Arithmetic, the expression \((p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}} \dots)\) runs through all positive integers exactly once. Thus upon re-arranging again we obtain the left hand of Euler’s formula.

The second proof, the one that Euler used, employs a sieve method. This time, we start with the left hand of the Euler product. If we multiply \(\zeta\) by \(2^{-s}\), we get back precisely the terms with \(n\) even. So

\[(1-2^{-s}) \zeta(s) = 1+3^{-s}+5^{-s}+\cdots = \sum_{2 \nmid n} n^{-s} \nonumber\]

Subsequently we multiply this expression by \((1-3^{-s})\). This has the effect of removing the terms that remain where \(n\) is a multiple of \(3\). It follows that eventually

\[(1-p_{l}^{-s}) \dots (1-p_{1}^{-s}) \zeta (s) = \sum_{p_{1} \nmid n, \cdots p_{l} \nmid n} n^{-s} \nonumber\]

The argument used in Eratosthenes sieve (Section 1.1) now serves to show that in the right hand side of the last equation all terms other than \(1\) disappear as l tends to infinity. Therefore, the left hand tends to 1, which implies the proposition.

The most important theorem concerning primes is probably the following (without proof).

Figure 3. On the left, the function \(\int_{2}^{x} \mbox{ln} t dt\) in blue, \(\pi (x)\) in red, and \(x/ \mbox{ln}x\) in green. On the right, we have \(\int_{2}^{x} \mbox{ln} t dt - x/\mbox{ln} x\) in blue, \(\pi (x)- x/\mbox{ln} x\) in red.

Theorem 2.21 (Prime Number Theorem)

Let \(\pi (x)\) denote the prime counting function, that is: the number of primes less than or equal to \(x > 2\).

• \(\lim_{x \rightarrow \infty} \frac{\pi (x)}{(x/\mbox{ln} x)} = 1\) and
• \(\lim_{x \rightarrow \infty} \frac{\pi (x)}{\int_{2}^{x} \mbox{ln} t dt} = 1\)

where \(\mbox{ln}\) is the natural logarithm.

The first estimate is the easier one to prove, the second is the more accurate one. In Figure 3 on the left, we plotted, for \(x \in [2,1000]\), from top to bottom the functions \(\int_{2}^{x} \mbox{ln} t dt\) in blue, \(\pi (x)\) in red, and \(x/\mbox{ln} x\). In the right hand figure, we augment the domain to \(x \in [2, 100000]\). and plot the difference of these functions with \(x/\mbox{ln} x\). It now becomes clear that \(\int_{2}^{x} \mbox{ln} t dt\) is indeed a much better approximation of \(\pi (x)\). From this figure one may be tempted to conclude that \(\int_{2}^{x} \mbox{ln} t dt - \pi (x)\) is always greater than or equal to zero. This, however, is false. It is known that there are infinitely many n for which \(\int_{2}^{x} \mbox{ln} t dt - \pi (x) < 0\). The first such \(n\) is called the Skewes number. Not much is known about this number, except that it is less than 10317.

Perhaps the most important open problem in all of mathematics is the following. It concerns the analytic continuation of \(\zeta (s)\) given above.

Conjecture 2.22 (Riemann Hypothesis)

All non-real zeros of \(\zeta (s)\) lie on the line \(\mbox{Re} s = \frac{1}{2}\)

In his only paper on number theory [ 20 ], Riemann realized that the hypothesis enabled him to describe detailed properties of the distribution of primes in terms of of the location of the non-real zero of \(\zeta ( s )\). This completely unexpected connection between so disparate fields – analytic functions and primes in \(\mathbb{N}-\)spoke to the imagination and led to an enormous interest in the subject. In further research, it has been shown that the hypothesis is also related to other areas of mathematics, such as, for example, the spacings between eigenvalues of random Hermitian matrices [ 2 ], and even physics [ 5 , 6 ].

The Riemann Hypothesis

A Resource for the Afficionado and Virtuoso Alike

• © 2008
• Peter Borwein 0 ,
• Stephen Choi 1 ,
• Brendan Rooney 2 ,
• Andrea Weirathmueller 3

Department of Mathematics & Statistics, Simon Fraser University, Burnaby, Canada

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Simon Fraser University, Burnaby, Canada

University of western ontario, fredericton, canada.

• Contains recent advances and results in number theory
• Collects papers never before published in book form
• Explains the Riemann Hypothesis to someone without a background in complex analysis

Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (12 chapters)

Front matter, introduction to the riemann hypothesis, why this book.

• Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller

Analytic Preliminaries

Algorithms for calculating ζ(s), empirical evidence, equivalent statements, extensions of the riemann hypothesis, assuming the riemann hypothesis and its extensions …, failed attempts at proof, original papers, expert witnesses, the experts speak for themselves, back matter.

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About this book

From the reviews:

"The Reimann Hypothesis presents itself as fundamentally a collection of well-known papers related to the Reimann Hypothesis, with a long introduction to set the stage. … This may be a useful resource for small libraries … and for those who might like to have copies of the papers in their personal library." (Fernando Q. Gouvêa, MathDL, January, 2008)

"This book is intended as a reference work on the Riemann Hypothesis (RH). … will undoubtedly be extremely useful for anyone making a serious study of the zeta-function, and especially those with an interest in the historical development of the subject. The choice of the material is good, and the discussion is helpful. … anyone working in the area will benefit from a study of them. Overall this is a book which belongs on the shelves of anyone interested in the RH." (Roger Heath-Brown, Zentralblatt MATH, Vol. 1132 (10), 2008)

"Borwein (Simon Fraser Univ.) and others have compiled mostly classic papers contributing to the theory of the distribution of prime numbers. … Summing Up: Recommended. Upper-division undergraduate through researchers/faculty." (D. V. Feldman, CHOICE, Vol. 45 (11), August, 2008)

"This delightfully written book on the Riemann Hypothesis is a welcome addition to the literature. … its structure makes it an ideal choice as a textbook for a reading course on the Riemann zeta function and its applications, especially in classes with students of diverse mathematical backgrounds and abilities. … I thoroughly enjoyed reading this book. … It is a great service to have them collected in one place, and this will increase the number of mathematicians who read them." (Steven Joel Miller, Mathematical Reviews, Issue 2009 k)

“This beautiful book is an in-depth introduction to the Riemann hypothesis, arguably the most famous unsolved problem of mathematics. … the book will also be of interest for anyone with an interest in thehistory of this result. … For everyone else it is a most valuable resource of information on a fascinating conjecture and a most welcome addition to the literature.” (C. Baxa, Monatshefte für Mathematik, Vol. 160 (3), June, 2010)

Editors and Affiliations

Department of mathematics & statistics, simon fraser university, burnaby, canada.

Peter Borwein, Stephen Choi

Brendan Rooney

Andrea Weirathmueller

Bibliographic Information

Book Title : The Riemann Hypothesis

Book Subtitle : A Resource for the Afficionado and Virtuoso Alike

Editors : Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller

Series Title : CMS Books in Mathematics

DOI : https://doi.org/10.1007/978-0-387-72126-2

Publisher : Springer New York, NY

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer-Verlag New York 2008

Hardcover ISBN : 978-0-387-72125-5 Published: 30 November 2007

Softcover ISBN : 978-1-4419-2465-0 Published: 23 November 2010

eBook ISBN : 978-0-387-72126-2 Published: 21 November 2007

Series ISSN : 1613-5237

Series E-ISSN : 2197-4152

Edition Number : 1

Number of Pages : XIV, 533

Topics : Number Theory , History of Mathematical Sciences

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Win a million dollars with maths, No. 1: The Riemann Hypothesis

The first million-dollar maths puzzle is called the Riemann Hypothesis . First proposed by Bernhard Riemann in 1859 it offers valuable insights into prime numbers but it is based on an unexplored mathematical landscape. If you can show that its mathematical path will always lie true, $1m (£600,000) is all yours.

Mathematicians are obsessed with primes because they are the foundation of all other numbers. Prime numbers in mathematics are like atoms in chemistry, bricks in the construction industry and ludicrous pay cheques in professional football. Everything is built up from these fundamental units and you can investigate the integrity of something by taking a close look at the units from which it is made. To investigate how a number behaves you look at its prime factors, for example 63 is 3 x 3 x 7. Primes do not have factors: they are as simple as numbers get.

They are simple in this one respect – but are otherwise extremely enigmatic and slip away just when you think you have a grip on them. Part of the problem is that, by definition, they have no factors, which is normally the first foothold in investigating a number problem. This is also the key to their usefulness. It is their difficulty to grasp that makes primes the basis for our modern information security. Whenever you use a cash machine or visit a secure website, it is huge prime numbers that encrypt your information and make it extremely difficult for anyone else to pry it back out of the electronic cipher.

Prime numbers also have the annoying habit of not following any pattern. 3,137 is a prime and the next one after that is not until 3,163, but then 3,167 and 3,169 suddenly appear in quick succession, followed by another gap until 3,187. If you find one prime number, there is no way to tell where the next one is without checking all the numbers as you go. One possible way to get a handle on how primes are spaced is to calculate, for any number, how many primes there are smaller than it. This is exactly what Riemann did in 1859: he found a formula that would calculate how many primes there are below any given threshold.

Riemann's formula is based on what are called the " Zeta Function zeroes ". The Zeta Function is a function that starts with any two coordinates and performs a set calculation on them to return a value. If you imagine the two initial coordinates to be values for latitude and longitude, for example, then the Zeta Function returns the altitude for every point, forming a kind of mathematical landscape full of hills and valleys. Riemann was exploring this landscape when he noticed that all of the locations that have zero altitude (points at "sea level" in our example) lie along an straight line with a "longitude" of 0.5 – which was completely unexpected. It's as if all the places in England that are at sea level (ignoring the coast) are on a dead straight line that runs directly north along the 0.5 longitude line.

Riemann used these zeroes as part of his prime distribution formula, but the problem is that no one knows for sure that all of the zeroes are on that same straight line. Sure, mathematicians have checked that the first ten trillion zeroes all fall on that line, but that's no guarantee that the ten trillionth and one zero might be somewhere else, throwing the whole prime distribution formula out the proverbial window, along with vast amounts of related number theory. Which is why there is a $1m prize for anyone who can show that all of the Zeta Function zeroes line up on the "0.5 line" without resorting to the impossible task of walking along this infinite line to check.

I've given you the Zeta Function to get you started and if you dust off a bit of "complex variable" maths, you will be well on your way to exploring the Riemann landscape. However – if that's a bit much – here is an easier starting problem: All prime numbers (greater than five) squared are one more than a multiple of 24. Check it for a few – it works. You can even prove that it works for all of the infinite number of primes.

Now if you can just do that for the Zeta zeroes, you can stop kicking a football around in the cold in hope of a big pay day.

Matt Parker is based in the mathematics department at Queen Mary, University of London, and can be found online at www.standupmaths.com His favourite prime is 31

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• 14 May 2024

Why mathematics is set to be revolutionized by AI

• Thomas Fink 0

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Giving birth to a conjecture — a proposition that is suspected to be true, but needs definitive proof — can feel to a mathematician like a moment of divine inspiration. Mathematical conjectures are not merely educated guesses. Formulating them requires a combination of genius, intuition and experience. Even a mathematician can struggle to explain their own discovery process. Yet, counter-intuitively, I think that this is the realm in which machine intelligence will initially be most transformative.

In 2017, researchers at the London Institute for Mathematical Sciences, of which I am director, began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they discovered that simple artificial intelligence (AI) classifiers can predict an elliptic curve’s rank 1 — a measure of its complexity. Elliptic curves are fundamental to number theory, and understanding their underlying statistics is a crucial step towards solving one of the seven Millennium Problems, which are selected by the Clay Mathematics Institute in Providence, Rhode Island, and carry a prize of US$1 million each. Few expected AI to make a dent in this high-stakes arena.

AI now beats humans at basic tasks — new benchmarks are needed, says major report

AI has made inroads in other areas, too. A few years ago, a computer program called the Ramanujan Machine produced new formulae for fundamental constants 2 , such as π and e . It did so by exhaustively searching through families of continued fractions — a fraction whose denominator is a number plus a fraction whose denominator is also a number plus a fraction and so on. Some of these conjectures have since been proved, whereas others remain open problems.

Another example pertains to knot theory, a branch of topology in which a hypothetical piece of string is tangled up before the ends are glued together. Researchers at Google DeepMind, based in London, trained a neural network on data for many different knots and discovered an unexpected relationship between their algebraic and geometric structures 3 .

How has AI made a difference in areas of mathematics in which human creativity was thought to be essential?

First, there are no coincidences in maths. In real-world experiments, false negatives and false positives abound. But in maths, a single counterexample leaves a conjecture dead in the water. For example, the Pólya conjecture states that most integers below any given integer have an odd number of prime factors. But in 1960, it was found that the conjecture does not hold for the number 906,180,359. In one fell swoop, the conjecture was falsified.

Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant. The On-Line Encyclopedia of Integer Sequences (OEIS) contains almost 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the formidable Busy Beaver sequence (0, 1, 4, 6, 13, …), which grows faster than any computable function. Scientists are already using machine-learning tools to search the OEIS database to find unanticipated relationships.

DeepMind AI outdoes human mathematicians on unsolved problem

AI can help us to spot patterns and form conjectures. But not all conjectures are created equal. They also need to advance our understanding of mathematics. In his 1940 essay A Mathematician’s Apology , G. H. Hardy explains that a good theorem “should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds”. In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us to reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing between them requires an intuition for how the field itself will evolve. This grasp of the broader context will remain out of AI’s reach for a long time — so the technology will struggle to spot important conjectures.

But despite the caveats, there are many upsides to wider adoption of AI tools in the maths community. AI can provide a decisive edge and open up new avenues for research.

Mainstream mathematics journals should also publish more conjectures. Some of the most significant problems in maths — such as Fermat’s Last Theorem, the Riemann hypothesis, Hilbert’s 23 problems and Ramanujan’s many identities — and countless less-famous conjectures have shaped the course of the field. Conjectures speed up research by pointing us in the right direction. Journal articles about conjectures, backed up by data or heuristic arguments, will accelerate discovery.

Last year, researchers at Google DeepMind predicted 2.2 million new crystal structures 4 . But it remains to be seen how many of these potential new materials are stable, can be synthesized and have practical applications. For now, this is largely a task for human researchers, who have a grasp of the broad context of materials science.

Similarly, the imagination and intuition of mathematicians will be required to make sense of the output of AI tools. Thus, AI will act only as a catalyst of human ingenuity, rather than a substitute for it.

Nature 629 , 505 (2024)

doi: https://doi.org/10.1038/d41586-024-01413-w

He, Y.-H., Lee, K.-H., Oliver, T. & Pozdnyakov, A. Preprint at arXiv https://doi.org/10.48550/arXiv.2204.10140 (2024).

Raayoni, G. et al. Nature 590 , 67–73 (2021).

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Davies, A. et al. Nature 600 , 70–74 (2021).

Merchant, A. et al. Nature 624 , 80–85 (2023).

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Riemann Hypothesis: What Yitang Zhang’s New Paper Means and Why You Should Care

A part of a visualisation of the Riemann zeta function in the complex plane. The Re( ½ ) line passes vertically through the middle. Plot: Nschlow/Wikimedia Commons, CC BY-SA 4.0

• Euclid found long ago that there are infinitely many prime numbers. However, they are not distributed evenly: they become less common as they become larger.
• The Riemann hypothesis makes an important statement about their distribution, offering to remove the seeming arbitrariness with which they turn up and impose order.
• The hypothesis is about the form that solutions to the Riemann zeta function, which could estimate the number of prime numbers between two numbers, are allowed to take.
• Yitang Zhang has claimed that he has disproved a weaker version of the Landau-Siegel zeroes conjecture, an important problem related to the hypothesis.
• The conjecture is that there are solutions to the zeta function that don’t assume the form prescribed by the Riemann hypothesis.

Earlier this October, Chinese websites claimed that the Chinese-American mathematician Yitang Zhang had solved the Landau-Seigel zeros conjecture – an important open problem in number theory related to cracking a bigger problem: is there a pattern to the way prime numbers are distributed on the number line?

It’s a simple question but it has only complicated answers. While mathematicians pursuing a resolution to the hypothesis may be motivated by their quest for knowledge alone, many others – including physicists – are interested because the answer has tantalising connections to many concepts in modern physics.

In 1859, the German mathematician Bernhard Riemann came close to answering the question when he formulated the Riemann hypothesis . It expressed an idea about a function he had discovered, called the Riemann zeta function, and its ability to estimate the number of prime numbers up to a particular point on the number line.

The Riemann hypothesis is often considered the most important unsolved problem in pure mathematics today. And Yitang Zhang has claimed that he has taken a big step towards solving it. Has he?

The zeta function

Prime numbers are the basic building blocks of natural numbers. Think of prime numbers as what atoms are to matter, or what alphabets are to a language. If you can create a periodic table of prime numbers, you will have a way to understand all numbers.

The Greek mathematician Euclid found long ago that there are infinitely many prime numbers. In the millennia since, we have learnt that 2 is the first prime[footnote]1 is neither prime nor composite[/footnote], 3 is the second prime, 5 is the third prime, 7 is the fourth prime, 11 is the fifth prime and so forth.

However, the prime numbers are not distributed evenly. They become less common as they become larger. For example:

* 40% of numbers up to 10 are prime numbers,

* 25% of numbers up to 100 are prime numbers,

* 16.8% of numbers up to 1,000 are prime numbers,

* 12.3% of numbers up to 10,000 are prime numbers,

* 9.6% of numbers up to 100,000 are prime numbers,

* 7.8% of numbers up to 1,000,000 are prime numbers,

… and so on.

But mathematicians don’t think this distribution is entirely arbitrary; they believe there could be a pattern. An important stop on the quest for this pattern is the prime number theorem.

Let’s start with the prime-counting function. Its graph (shown below) is that of a counter: flat when there is no prime number, +1 when there is a prime number. It is denoted as π( x ). x here is the position on the real number line. The fraction of prime numbers up to x is π( x )/ x .

In the late 1700s, when he was still a teenager, the German mathematician Carl Friedrich Gauss had an idea: that for large values of x , the value of the prime counting function π( x ) will be approximately equal to that of the logarithmic integral function. In other words, at a point far along the number line, the total number of prime numbers until that point will get closer and closer to the value of the function shown below:

In yet other words, at position x , the ratio of the number of prime numbers to x will be roughly equal to 1/log x .

This is the prime number theorem as formulated by Gauss.

In the 1850s, Riemann – who was a student of Gauss – investigated Gauss’s conjecture. He discovered a profound connection between the conjecture and the zeta function – another function that had been investigated by the famous Swiss mathematician Leonard Euler a century earlier.

The zeta function is an infinite sum of the following form:

Euler had proved that when s > 1, the sum ζ( s ) has a finite value. He found that ζ(2) is equal to π 2 /6…

… and that ζ(4) is equal to π 4 /90.

Euler also found that the zeta function, which is expressed as an infinite sum , could be expressed as an infinite product :

The Riemann hypothesis

Riemann studied the zeta function using a branch of mathematics he pioneered called complex analysis. Specifically, he used a technique called analytic continuation to make sense of the values of the zeta function for complex inputs.

That is, he found a way to calculate the value of ζ( s ) when s is a complex number.

A complex number is any number of the form a + b i . Here, a and b are real numbers and i is the imaginary unit: i = √-1.

Riemann observed that in the new domain of complex numbers, for some values of s , the value of ζ( s ) was 0. These values of s are called the zeta zeroes. Some of them were easier to find. For example, for every negative even integer s (-2, -4, -6, …), ζ( s ) equals 0. Riemann called these the trivial zeros . (-2 is not a complex number but can be expressed as one: -2 + 0 i .)

There are other zeta zeroes called the non-trivial zeroes – they form the crux of the Riemann hypothesis. Riemann could show that on a graph, all the non-trivial zeroes should lie in a region called the critical strip – between the vertical lines passing through 0 and 1 on the x-axis (see below).

The Riemann hypothesis is the statement that all the non-trivial zeroes should lie not just somewhere in the strip but on a single vertical line called the critical line , which passes through 1/2 on the x-axis.

In technical terms

Recall that Riemann’s first insight was a connection between Gauss’s conjecture and the zeta function. Gauss’s conjecture stated that the value of π( x ) for a large x is roughly equal to the value of a different function.

Riemann found this connection when he modified the prime-counting function π( x ) and arrived at a new formula, J ( x ):

The first term, Li ( x ), approximates Gauss’s original prime counting function, π( x ). ‘Li’ stands for ‘logarithmic integral’. The second term is the sum of the logarithmic integral of x to the power ρ [footnote]Pronounced ‘rho’[/footnote], and summed over ρ . The ρ denotes the non-trivial zeroes of the zeta function. The term c is a constant.

Say the non-trivial zeroes are the complex numbers a , b , c , d and e . The sum will then be over a to e : ( Li ( x a )) + ( Li ( x b )) + ( Li ( x c )) + ( Li ( x d )) + ( Li ( x e )), and finally with the addition of the constant.

The more non-trivial zeroes the function sums over, the tighter its estimate of the prime count will be. That is, the estimate after summing over a thousand zeroes will be better than the estimate after summing over a hundred zeroes.

Also read: Beyond the Surface of Einstein’s Relativity Lay Riemann’s Chimerical Geometry

The form of this function revealed to Riemann that the locations of prime numbers are deeply connected to the locations of the non-trivial zeroes of the zeta function. More specifically, Riemann used the formula to hypothesise that the complex-number representation of each non-trivial zero always had the form:

½ + <a real number> times i

In technical parlance, the Riemann hypothesis states that “the nontrivial zeros of ζ( s ) lie on the line Re( s ) = ½”.

Jacques Hadamard and Charles Poussin proved Gauss’s conjecture independently in the 1890s using Riemann’s work on the zeta function. Their result is now called the prime number theorem (which we encountered earlier). However, Riemann’s hypothesis itself continues to resist attempts to prove to this day.

Landau-Siegel zeroes

An arithmetic progression is a sequence of numbers where the next term is the previous term plus a constant. For example, 1, 3, 5, 7, 9… is an arithmetic progression where the constant is +2.

In 1837, the German mathematician P.G.L. Dirichlet proved that there are infinitely many prime numbers in certain arithmetic progressions. He was able to do so by using a modified version of the zeta function, known today as the Dirichlet L-function . It is effectively a generalised form of the zeta function. It looks like this:

Here, s is a complex number and Χ [footnote]Pronounced ‘chi’[/footnote] is a function that takes natural numbers as inputs and spits out complex numbers. (‘Σ’ is the summation symbol.)

The generalised Riemann hypothesis (GRH) is the conjecture that all zeros of the L-function L ( s , Χ ) that lie in the critical strip should also lie on the critical line. It’s the same hypothesis as before, reapplied to the L-function.

Now, a Landau–Siegel zero of the function L ( s , Χ ) is any real number between ½ and 1 that, when used for s , makes L ( s , Χ ) equal 0.

It is in effect a counterexample to the GRH: it implies that there could be a real number in the critical strip that doesn’t lie on the critical line, yet is a zero of the generalised zeta function.

Obviously, proving that there are no Landau-Siegel zeroes would be a weak form of proving the GRH. It wouldn’t allow us to claim that all non-trivial zeroes of the generalised function lie on the critical line – but it would allow us to say that some non-trivial zeroes definitely don’t lie outside the line.

Yitang Zhang’s preprint

On November 7, the Chinese-American mathematician Yitang Zhang announced that he had achieved a breakthrough in the study of Landau-Siegel zeroes. Specifically, he claimed to have proved a weaker version of the Landau-Siegel zeroes conjecture.

Note that this weakness is in addition to the weakness of the conjecture relative to the GRH – so the claim is in effect doubly weak. That said, it holds the hope of taking us closer to a highly complex and longstanding problem, so it merits our attention and scrutiny.

As number theorist Alex Kontorovich told Nature , “Resolving any of these issues would be a major advancement in our understanding of the distribution of prime numbers.”

Disproving the existence of Landau-Siegel zeroes requires mathematicians to prove that L (1, Χ ) is much greater than (log D ) -1 .

In 2007, Zhang had published a preprint paper claiming that he had proved that L(1, Χ ) was much greater than (log D ) -17 (log(log D )) -1 . But his proof turned out to be wrong after mathematicians noticed the incorrectness of a few key ideas developed in that paper.

In his new paper, Zhang claims to have proved that L(1, Χ ) is much greater than (log D ) -2022 . He wrote about his work on a Chinese website called Zhihu , where he appears to claim that he tweaked his calculations to achieve the exponent to be 2022 – the year in which the result has been announced.

Zhang laboured in obscurity before he shot to fame in 2013 for his work on the twin-prime conjecture. He proved that there are infinitely many pairs of prime numbers separated by an even number that is lower than 70,000,000. The original conjecture is that there are infinitely many primes separated by an even number equal to 2. Zhang’s achievement was to bring this constant down from ‘finite but large’ to below 70 million.

The result was considered a great advancement. The Fields Medal winner Terence Tao initiated a large collaborative project called PolyMath8 to improve Zhang’s techniques and to lower the bound from 70 million. The 2022 Fields medallist James Maynard obtained a different proof of Zhang’s theorem which resulted in the project’s successful completion by lowering the bound to just 246.

It appears that Zhang did not use the ideas in his 2007 article in his new paper. Other mathematicians are only just beginning to review his work. Zhang has also said that his new techniques can be improved to lower the value of the exponent to the hundreds.

If Zhang’s claim is established, there is a chance that there will be another large PolyMath project to improve Zhang’s techniques. Even bringing the exponent down to the hundreds – as Zhang has said might be possible – won’t prove the Landau-Seigel zeroes conjecture but will be a significant advancement in the annals of number theory.

The reason is that there are several conjectures in fields as diverse as cryptography and quantum physics whose framing depends on the validity of the GRH. If GRH is proved, it will immediately also establish the correctness of all these other conjectures.

In many of these conjectures, the hypothesis can be replaced by a weak form of the Landau-Seigel conjecture of the type L (1, X ) >> (log D) – n , where n is any finite number. So if Zhang’s new result, with the -2022 exponent, is true, it will right away also prove these other conjectures.

Potential implications

One field where a resolution of the Riemann hypothesis will have a large effect is modern cryptography. A common encryption method involves an encryption key, which is public, and a decryption key, which is kept private. The decryption key is composed of two large prime numbers, and the encryption key is the product of these two numbers. Anyone can encrypt a message using the encryption key, but only the person holding the decryption key can decode it.

If the Riemann hypothesis is proved, it could lead to new techniques to find large prime numbers. This in turn could ease methods to factorise the encryption key into its two prime numbers, which would reveal the decryption key. Thus, cryptographers will have  to find a new way to secure information – one that doesn’t depend on prime-number factorisation – as system administrators scramble to secure sensitive data like user passwords, banking data, etc.

A solution to the Riemann hypothesis is also expected to open up beneficial applications. Quantum chaos is a subfield of quantum physics that studies quantum systems whose classical counterparts exhibit chaotic behaviour.

To borrow an example that Barry Cipra used in a 1998 article : It is easy to predict the path of a billiards ball rolling around in a rectangular space – and it’s equally easy when the ball is replaced with an electron. But when the ball is set in motion inside a pill-shaped box, its path becomes chaotic. Quantum chaos is concerned with predicting the path when the ball is replaced by an electron in the second case.

It uses a class of equations called trace equations – and it so happens that the Riemann zeta function has the form of a trace equation. This means a resolution to the Riemann hypothesis can help physicists create a quantum-chaotic system, like the electron in a pill-shaped box, and thus bring more order to the subfield and its widespread complexity.

A third potential implication (among several) is that the spacing of the zeroes of the Riemann zeta function roughly resembles the spacing between the energy levels of a heavy nucleus, such as erbium-166. Say you mark the energy of each of these levels as points on the number line. Then you derive the point-correlation function: it’s a function that tells you how many points on the line are separated by a distance d , where you get to pick d .

In 1972, the American mathematician Hugh Montgomery and the British physicist Freeman Dyson found that the pair-correlation function of the zeroes of the Riemann zeta function resembled the pair-correlation function used to describe the energy levels of a heavy nucleus.

This points to a deep connection between the Riemann zeta function and nuclear physics – and one that is emblematic of similar connections between the function and patterns in many branches of modern physics.

Mathematicians and physicists have interpreted these links to mean that the Riemann hypothesis ought to be true. But the only way Riemann’s conjecture can be proven once and for all is using the techniques of number theory. To this end, the new paper by Yitang Zhang offers another way forward.

With inputs from Vasudevan Mukunth.

Mohan R. is an assistant professor at Azim Premji University Bangalore. He works in the areas of mathematics communication and mathematics education.

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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics

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Karl Sabbagh , Jonathan P. Keating; The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. Physics Today 1 June 2004; 57 (6): 63. https://doi.org/10.1063/1.1784280

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Mathematics > Number Theory

Title: values of random polynomials under the riemann hypothesis.

Abstract: The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitraliy small $\varepsilon>0$. Usually, it is more difficult to obtain similar order of error term for a higher degree polynomial $f(x)$ in place of $\text{Id}(x)$. Under the Riemann hypothesis, we show that the error term, on average in a weak sense, over polynomials of arbitrary degree, is of the expected order $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$.

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Famed mathematician claims proof of 160-year-old Riemann hypothesis

By Gilead Amit

21 September 2018

Michael Atiyah claims to have found a proof of the Riemann hypothesis

James Glossop/The Times/News Syndication

One of the most important unsolved problems in mathematics may have been solved, retired mathematician Michael Atiyah is set to claim on Monday. In a talk at the Heidelberg Laureate Forum in Germany, Atiyah will present what he refers to as a “simple proof” of the Riemann hypothesis , a problem which has eluded mathematicians for almost 160 years.

Born in 1929, Atiyah is one of the UK’s most eminent mathematical figures, having received the two awards often referred to as the Nobel prizes of mathematics, the Fields medal and the Abel Prize. He also, at various times, served as president of the London Mathematical Society, the Royal Society and the Royal Society of Edinburgh.

If a solution to the Riemann hypothesis is confirmed, it would be big news. Among other things, the hypothesis is intimately connected to the distribution of prime numbers , those indivisible by any whole number other than themselves and one. If the hypothesis is proven to be correct, mathematicians would be armed with a map to the location of all such prime numbers, a breakthrough with far-reaching repercussions in the field.

Read more: Mathematicians shocked to find pattern in ‘random’ prime numbers

As one of the six unsolved Clay Millennium Problems , any solution would also be eligible for a $1 million prize. The prestige has tempted many mathematicians over the years , none of which has yet been awarded the prize.

Atiyah is well aware of this history of failure. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he says, but he hopes his presentation will convince his critics.

In it, he pays tribute to the work of two great 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, whose developments he claims laid the foundations for his own proposed proof. “It fell into my lap, I had to pick it up,” he says.

New Scientist  contacted a number of mathematicians to comment on the claimed proof, but all of them declined. Atiyah has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers.

“People say ‘we know mathematicians do all their best work before they’re 40’”, says Atiyah. “I’m trying to show them that they’re wrong. That I can do something when I’m 90.”

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