riemann hypothesis math problem

July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

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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Riemann hypothesis

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Riemann hypothesis , in number theory , hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function , which is connected to the prime number theorem and has important implications for the distribution of prime numbers . Riemann included the hypothesis in a paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (“On the Number of Prime Numbers Less Than a Given Quantity”), published in the November 1859 edition of Monatsberichte der Berliner Akademie (“Monthly Review of the Berlin Academy”).

Riemann extended the study of the zeta function to include the complex numbers x + i y , where i = Square root of √ −1 , except for the line x = 1 in the complex plane. Riemann knew that the zeta function equals zero for all negative even integers −2, −4, −6,… (so-called trivial zeros) and that it has an infinite number of zeros in the critical strip of complex numbers that fall strictly between the lines x = 0 and x = 1. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1 / 2 . Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.

In 1914 English mathematician Godfrey Harold Hardy proved that an infinite number of solutions of ζ( s ) = 0 exist on the critical line x = 1 / 2 . Subsequently it was shown by various mathematicians that a large proportion of the solutions must lie on the critical line, though the frequent “proofs” that all the nontrivial solutions are on it have been flawed. Computers have also been used to test solutions, with the first 10 trillion nontrivial solutions shown to lie on the critical line.

A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography .

The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics . It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated Hilbert’s idea with a list of important problems for the 21st century; the Riemann hypothesis was number one. In 2000 it was designated a Millennium Problem , one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., U.S., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency ( DARPA ) listed it as one of the DARPA Mathematical Challenges, 23 mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Nineteen: Settle the Riemann Hypothesis. The Holy Grail of number theory.”

The Riemann Hypothesis: A 160-Year-Old, Million-Dollar Maths Problem

Since its publication in 1859, the riemann hypothesis has become one of the holy grail of mathematics..

Christopher McFadden

The Riemann Hypothesis is one of the most important mathematical advancements in history. Devised in by  Georg Friedrich Bernhard Riemann in  1859 it has yet to be rivaled in its impact, or solved, for almost 160 years. 

That is, until now – if a current claim of its proof is true. 

But what is it? Why is it important? Let’s take a quick look at this keystone of modern prime number theorem.

What is the Riemann Hypothesis? 

The Riemann Hypothesis was a groundbreaking piece of mathematical conjecture published in a famous paper  Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse ( “On prime numbers less than a given magnitude”) in 1859 by Bernhard Riemann.

It centers around one of the most fascinating phenomena of maths – prime numbers.  Mathematicians have been on a quest to predict them since the discipline’s very beginnings.

Maths ‘giants’ like Euclid, Euler’s product formula (connecting prime numbers to the zeta function), Gauss (Riemann was a student of Gauss) and Legendre (formulation of the prime number theorem) to Hadamard and de la Vallée Poussin have all made huge contributions to the field over the centuries. 

Prime numbers tend to not follow any discernable pattern. By finding one you cannot predict the next one without studying other numbers as you move forward – far from an efficient process.

Some had postulated that instead of looking forward, it might be useful to look backward instead. Could it be possible to understand the spacing of prime numbers by seeing how many there are smaller than the current one?

This is exactly what Riemann attempted to achieve. This insight would result in  Riemann making one of the biggest leaps in our understanding of prime number theory since antiquity. Not just that but for almost 160 years  it is a feat yet to be matched or exceeded. 

The Riemann hypothesis centers around the zeta function equation

As the Clay Mathematics Institute explains:

“[Riemann] observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function:-

ζ(s) = 1 + 1/2 s  + 1/3 s  + 1/4 s  + …

[This is] called the  Riemann Zeta function . The Riemann hypothesis asserts that all  interesting  solutions of the equation:-

lie on a certain vertical straight line.”

In overly simplified terms it relates to the distribution of prime numbers, but that doesn’t really begin to explain it.  A more in-depth explanation (which is needed) of this is out of the scope of this article but  Jørgen Veisdal (a Ph.D. fellow at the  Norwegian University of Science and Technology) has made a very informative overview.

His work now forms the main focus of prime number theory and was the main reason for the proof of prime number theorem in 1896 .  Since then several new proofs have been found, including elementary proofs by Selberg and Erdós. Riemann’s hypothesis about the roots of the zeta function, however, remains a mystery.

Although very complex in nature, what it is trying to solve is pretty simple. Instead of trying to determine where prime numbers were, Riemann attempted to investigate the very nature of them. 

He was not the first to take this approach, in fact, it was the ‘fashion’ of his peers during the 1800’s but he was to become a master of it. 

Why is the Riemann Hypothesis important?

In short, it’s sort of a ‘holy grail’ of mathematics. “Most mathematicians would trade their soul with Mephistopheles for a proof,” said Marcus du Sautoy of Oxford University.

To date, mathematicians have a pretty good idea, approximations, for the density of primes but not to an absolute certainty. These approximations are just that and no function (yet known) exists that allows them to efficiently and perfectly compute the number of primes less than a given integer (which tend to be numbers with millions of zeros).

Given that mathematicians are unable to determine exact values, they want to know how good their approximations are. This is the problem Reimann was trying to address with his  1859 paper.

If his hypothesis is true it would guarantee a far greater bound on the difference between existing approximations and the ‘real’ value. It would, in other words, tell us if prime numbers are as chaotic as they seem today.

Although his hypothesis does tackle hundreds of other concepts, its core is concerned with the distribution of prime numbers.

To some this might seem like a “lot of fuss about nothing” but when you realize that many large organizations, like the NSA, recruit many number theorists to conduct research in this field something is definitely important about it.

Prime number theorem was once purely theoretical but has started to find real-world applications in our modern digital world. cell-phones, for example, would not be able to work without spread spectrum communications and “quadratic residue sequences”.

Both these rely heavily on some properties of prime numbers to allow multiple signals to work on the same frequency band. 

But more importantly, prime number factorization is a commonly employed practice used in encryption techniques like public-key encryption systems. These tend to use large semi-primes (multiplying two prime numbers) to secure the encryption. 

In order to break it, you would have to find the prime factorization of the large semi-prime number – that is, two or more prime numbers that multiplied together result in the original number.

When this technique employs small prime numbers it’s relatively simple to crack but using larger ones can take even computers days, months even years to solve. Given the non-linear distribution of prime numbers, the process is a trial and error process – you’ll need to try all the possible combinations. 

In short, solving it would, amongst other things, have enormous implications for cyber-security .  

The Riemann hypothesis is one of the Millennium Prize Problems

There are some problems that have remained stubbornly beyond the abilities of our greatest minds. Some of these, at least in the field of mathematics, is called the Millenium Prize Problems . 

These consist of a seven (now six) or so mathematical problems that were identified by the Clay Mathematics Institute at the turn of the new millennium.

To date, they comprise of the following:-

1. Yang-Mills and Mass Gap  – According to quantum solutions of Yang-Mills equations there appears to be a “mass gap”. 

2. Riemann Hypothesis  

3. P versus NP Problem – This is typified by the Hamilton Path Problem. “G iven N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.”

4.  Navier–Stokes Equation  – The equation that governs the flow of fluids. “However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique?” It has been claimed,  though not officially recognized by the Clay Mathematical Institute, that this has been solved by Mukhtarbay Otelbayev. 

5. Hodge Conjecture – “ The answer to this conjecture determines h ow much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations.” This problem asks whether complex mathematical shapes can be built from simple ones. 

6.  Poincaré Conjecture – French mathematician Henri Poincare asked, in 1904 , if the three-dimensional sphere is characterized as “t he unique simply connected three manifold.” It is a special case of Thurston’s geometrization conjecture . 

7.  Birch and Swinnerton-Dyer Conjecture – “ Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational point”. This is also as one of the most challenging mathematical problems still to be solved.

How many Millennium Problems have been solved?

These problems each carry an eye-watering $1 million cash prize but the real award is the ever-lasting fame and respect from peers to come along with solving them. 

To date, only one of the original seven has been solved. Officially only the  Poincaré Conjecture has been solved.  This was solved by Russian mathematician Grigori Perelman in 2003 . 

Perelman has built a career out of solving mathematical problems and has made significant contributions to  Riemannian geometry  and  geometric topology . In 2006 he was honored in the December 22nd edition of Nature for his solution to the Poincare’s conjecture marking it as the scientific “Breakthrough of the Year”. 

When it was officially announced he met the criteria for the Clay Millenium Prize in 2010 he rejected the prize money stating that his contributions were no greater than those of Richard S. Hamilton. 

But the Riemann hypothesis might be the next one to fall if the recent news turns out to be correct.  It seems a 90-year-old retired mathematician may have a solution that has plagued his peers for almost 160 years . 

Of course, his claim will need to be verified by the Clay Mathematical Institute first, but it could mean  the Riemann hypothesis has finally been solved.

But he is not the first to claim to have solved the  Reimann hypothesis. In 2004,  Louis de Branges, a French-born mathematician, now at Purdue University in the US, claimed a proof of the Riemann hypothesis. 

Branges’ solution was, however, later dismissed by his peers .  

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Riemann Hypothesis

Some equivalent statements of the Riemann Hypothesis are

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

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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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The Riemann hypothesis, a Step Closer to Being Solved

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The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved Scientific American Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed … link

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 BCE that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line….

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The Riemann Hypothesis Explained

This is quite a complex topic probably only accessible for high achieving HL IB students, but nevertheless it’s still a fascinating introduction to one of the most important (and valuable) unsolved problems in pure mathematics.

Firstly, the Riemann Hypothesis is concerned with the Riemann zeta function.  This function is defined in many ways, but probably the most useful for us is this version:

In other words the Riemann zeta function consists of a sum to infinity multiplied by an external bracket.  s is a complex number of the form s  =  σ  +  it.   This formula is valid for Re(s) > 0 .  This means that the real part of the complex number must be positive.

Now, the Riemann Hypothesis is concerned with finding the roots of the Riemann zeta function – ie. what values of complex number s cause the function to be zero.  However the equation above is only valid for Re(s) > 0.  To check for roots where Re(s) is less than or equal to 0 we can use an alternative representation of the Riemann zeta function:

which shows that the zeta function is zero whenever s = -2,-4,-6…. as for these values the sine term becomes zero.  (s = 0 has no solution in this representation because it leaves us with a zeta function of 1 in the far right term – which produces a singularity).  These values are called the trivial zeroes of the zeta function.

The other, non-trivial zeroes of the Riemann zeta function are more difficult to find – and the search for them leads to the Riemann hypothesis:

The non-trivial zeroes of the Riemann zeta function have a real part of s equal to 1/2

In other words, ζ ( s ) has non-trivial zeroes only when s is in the form s = 1/2 + it.  This is probably easier to understand in graphical form.  Below we have s plotted in the complex plane:

We can see that when s = 1 the function is not defined.  This is because when s =1 in the original equation for the zeta function we get a singularity as this causes the bracket to the left of the summation to reduce to 1/0.    All the non-trivial zeroes for the zeta function are known to lie in the grey boxed, “critical strip” – and the Riemann hypothesis is that they all lie on the dotted line where the real value is 1/2.

This hypothesis, made by German mathematician Bernhard Riemann in 1859 is still unsolved over 150 years later – despite some of the greatest mathematical minds of the 2oth century attempting the problem.  Indeed it is considered by many mathematicians to be the most important unresolved question in pure mathematics.  Mathematician David Hilbert who himself collected 23 great unsolved mathematical problems together in 1900 stated,

“If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?”

The problem is today listed as one of the Clay Institute’s Millennium Prize Problems – anyone who can prove it will win $1 million and will quite probably go down in history as one of the greatest mathematicians of all time.

One solution for s which gives a zero of the zeta function is 0.5 + 14.134725142 i.  Another one is 0.5 + 21.022039639 i.  These both satisfy the Riemann Hypothesis by having a real part of 1/2.  Indeed, to date, 10 trillion (10,000,000,000,000) non trivial solutions have been found – and they all have a real part of 1/2.  But this is not a proof that it is true for all roots  – and so the problem remains unsolved.

So, why is this such an important problem?  Well, because there is a connection between the Riemann zeta function and distribution of prime numbers.  The function below on the left is another way of representing the Riemann zeta function and the function on the right is an infinite product including all prime numbers:

Understanding the Riemann zeta function will help mathematicians unlock some of the mysteries of the prime numbers – which are the building blocks of number theory (the study of integers).  For example looking at the graph below (drawn by a Wolfram Mathlab probject) we can see the function pi(x) plotted against a function which uses both the Riemann zeta function and the distribution of its zeroes.  pi(x) is blue graph and shows the number of primes less than or equal to x.

If you are interested in reading more on this the Wikipedia page on the Riemann zeta function  goes into a lot more detail.  A more lighthearted introduction to the topic is given by the paper, “ A Friendly Introduction to the Riemann Hypothesis ”

If you enjoyed this post you might also like:

Graham’s Number – a number so large it’ll literally collapse your head into a black hole.

Twin Primes and How Prime Numbers are Distributed – some more discussion on studying prime numbers – in particular the conjecture that there are infinitely many twin primes.

The Million Dollar Maths Problems – some general introductions to the seven million dollar maths problems.

Essential resources for IB students:

1) Exploration Guides  and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here  and the Paper 3 Questions can be downloaded here .

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4 thoughts on “ the riemann hypothesis explained ”.

There is an error in the first expression of Zeta(s), it should be (-1)^(n+1) not (-1)^n.

-1+0 (raised to the power of 0)/2 = 1/2 + 0. QED.TM.RH.

i^2+0^0/2=0+1/2i. i^2+(x+y+z)/2=0+1/2r. QED. RH. PROVED!! TRUE!!

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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a solution have?

Many researchers have spent a lot of time on it, trying to prove it, but why is it important to solve the problem?

I've tried relating the situation to problems in my field. For instance, solving the $P \ vs. NP$ problem has important implications if $P = NP$ is shown, and important implications if $P \neq NP$ is shown. For instance, there would be implications regarding the robustness or security of cryptographic protocols and algorithms. However, it's hard to say WHY the Riemann Hypothesis is important.

Given that the Poincaré Conjecture has been resolved, perhaps a hint about what to expect if and when the Riemann Hypothesis is resolved could be obtained by seeing what a proof of the Poincaré Conjecture has led to.

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• 7 $\begingroup$ Define "practical use". And it is important because it is there and it is fun to know stuff, though this would probably not fit within the "practical use" tag... $\endgroup$ –  DonAntonio Commented May 28, 2013 at 9:51
• $\begingroup$ It is very likely to be a duplicate isn't it? $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 9:52
• $\begingroup$ @DominicMichaelis; I don't follow, a duplicate of what? $\endgroup$ –  Mythio Commented May 28, 2013 at 10:00
• 4 $\begingroup$ Well it is a very famous hypothesis so you won't be the first one to ask it, partial answers are for example here $\endgroup$ –  Dominic Michaelis Commented May 28, 2013 at 10:14
• 3 $\begingroup$ immortality ... in the sense of Pythagoras ($a^2 + b^2 = c^2$) and Einstein ($E=mc^2$) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:43

5 Answers 5

Proving the Riemann Hypothesis will get you tenure, pretty much anywhere you want it.

• 16 $\begingroup$ Worth the downvotes :) $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:51
• 16 $\begingroup$ Not to mention the 1M on your bank account... $\endgroup$ –  Matemáticos Chibchas Commented Dec 29, 2013 at 3:44
• 6 $\begingroup$ I've already proven it, I just need to check my calculations... ;) $\endgroup$ –  King Squirrel Commented May 4, 2014 at 17:26
• 37 $\begingroup$ @KingSquirrel If only that damn margin were to be a bit wider... ;-) $\endgroup$ –  triple_sec Commented Aug 26, 2014 at 7:46
• 1 $\begingroup$ @KingSquirrel I have proved it too, just waiting for the prize to be increased to $10M.;) $\endgroup$ –  ishandutta2007 Commented Aug 12, 2022 at 8:55

The Millennium problems are not necessarily problems whose solution will lead to curing cancer. These are problems in mathematics and were chosen for their importance in mathematics rather for their potential in applications.

There are plenty of important open problems in mathematics, and the Clay Institute had to narrow it down to seven. Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis. There are many (many many) theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true.

In contrast, a solution to some of the other Millennium problems is (highly likely) not going to lead to anything dramatic. For instance, the $P$ vs. $NP$ problem. I personally doubt it is probable that $P=NP$ . The reason it's an important question is not because we don't (philosophically) already know the answer, but rather that we don't have a bloody clue how to prove it. It means that there are fundamental issues in computability (which is a hell of an important subject these days) that we just don't understand. Solving $P \ne NP$ will be important not for the result but for the techniques that will be used. (Of course, in the unlikely event that $P=NP$ , enormous consequences will follow. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.)

The Poincaré conjecture is an extremely basic problem about three-dimensional space. I think three-dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three-dimensional space, but I'm sure there are.

• 16 $\begingroup$ about Perelman's solution and the Poincaré conjecture : in theoretical physics, more precisely in general relativity, there are equations relating the distribution of mass in the universe to the curvature of space-time. Perelman's solution used curvature (via the Ricci flow) to get topological information, i.e. the shape of the compact simply connected manifold (homeomorphic to a sphere). these kind of techniques are related to determnining the shape of the universe (space-time) which is a 4-manifold, depending on the curvature (which is determined by mass repartition, which we can observe) $\endgroup$ –  Albert Commented May 28, 2013 at 13:22
• 2 $\begingroup$ @Glougloubarbaki - it would be great if you could expound your comment as an answer. $\endgroup$ –  Nathaniel Bubis Commented Dec 29, 2013 at 2:45
• 2 $\begingroup$ Is it really obvious that "enormous consequences will follow" from a proof of P = NP? $\endgroup$ –  bof Commented Nov 14, 2014 at 4:53
• $\begingroup$ @bof highly likely, yes. At the very least lots and lots of expert computer scientists will look at the proof in disbelief as it shatters something they really felt very confident about. And, unless the proof will be completely unusable computationally, some really hard to solve problems will be found polynomial solutions. $\endgroup$ –  Ittay Weiss Commented Nov 14, 2014 at 4:57
• 1 $\begingroup$ P=NP. I'd love an opportunity to demonstrate my NTM. Nonetheless, the argument that no-one believes P=NP is just patently false. P=NP via BPP. $\endgroup$ –  Jonathan Charlton Commented Jan 18, 2016 at 6:44

Explaining the true mathematics behind the Riemann Hypothesis requires more text that I'm allotted (took most of my undergraduate degree in mathematics to even touch the surface; required all of graduate school to fully appreciate the beauty).

In very simple terms, the Riemann Hypothesis is mostly about the distribution of prime numbers. The idea is that mathematicians have some very good approximations (emphasis on approximate) for the density of the primes (so you give me an integer, and I can use these approximate functions to tell you roughly how many primes are between 0 [really 2] and that integer). The reason we use these approximations is that no [known] function exists that efficiently and perfectly computes the number of primes less than a given integer (we're talking numbers with literally millions of zeros). Since we can't determine the exact values (again, I'm simplifying a lot of this) the problem mathematicians want to know is exactly HOW good are these approximations.

This is where the Riemann Hypothesis comes in to play. For well over a century, mathematicians have known that a special form of the polylogarithm function (again, more fun math if you're bored) is a really great approximation for the prime counting function (and it's way easier to compute). The Riemann Hypothesis, if true, would guarantee a far greater bound on the difference between this approximation and the real value. In other words, the importance of the Riemann Hypothesis is that it tells us a lot about how chaotic the primes numbers really are. That's an incredibly high-level explanation and the Riemann Hypothesis deals with literally hundreds of other concepts, but the main point is understanding the distribution of the primes.

The Riemann hypothesis is a conjecture about the Riemann zeta function $$\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s}$$ This is a function $\mathbb{C} \rightarrow \mathbb{C}$. With the definition I have provided the zeta function is only defined for $\Re(s)\gt1$. With some complex analysis you can proof that there is a continuous (actually holomorphic if you know what it means) extension of the function so that it is defined in whole $\mathbb{C}$. The Riemann zeta function has some trivial zero points like $-2,-4,-6.$ The hypothesis says that the other zero points lie on the critical line $\Re(s)=\dfrac12$. This hypothesis had many application in analysis and number theory. The first proof of the prime number theorem used this conjecture.

In order to give an anwer to your question a would like to refer to this website, where you can find tons of applications of the Riemann hypothesis.

• 38 $\begingroup$ This is not an answer to the question, but a (very) brief introduction to what the RH is. OP asked about what applications proving RH would have. $\endgroup$ –  Lord_Farin Commented May 28, 2013 at 9:58
• 2 $\begingroup$ the link was interesting (+1) $\endgroup$ –  zerosofthezeta Commented Sep 14, 2013 at 7:46
• $\begingroup$ And we want the zeros of $\zeta(s)$ because $\frac{\zeta'(s)}{s\zeta(s)}$ is the Laplace transform of $\Psi(u) = \sum_{p^k < e^u} \log p$ a function showing us the distribution of primes numbers. If the RH is true, then $\frac{\Psi(u) -e^u}{e^{u/2}}= o(e^{-u/2})+\log 2\pi + \sum_\gamma \frac{e^{i \gamma u}}{1/2+i\gamma}$ where $\gamma$ are the imaginary parts of the non-trivial zeros $\endgroup$ –  reuns Commented Dec 23, 2016 at 7:33

The techniques used in the proofs of some of the most difficult theorems are used to prove so many other theorems. A proof of 1 of these theorems will give us access to an incredible amount of new techniques that will definitely make mathematics shorter,simpler and easier to understand.

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How I Learned to Love and Fear the Riemann Hypothesis

January 4, 2021

Emily Buder/Quanta Magazine; Guan-Huei Wu and Clay Shonkwiler for Quanta Magazine

Introduction

I first heard of the Riemann hypothesis — arguably the most important and notorious unsolved problem in all of mathematics — from the late, great Eli Stein , a world-renowned mathematician at Princeton University. I was very fortunate that Professor Stein decided to reimagine the undergraduate analysis sequence during my sophomore year of college, in the spring of 2000. He did this by writing a rich, broad set of now-famous books on the subject (co-authored with then-graduate student Rami Shakarchi).

In mathematics, analysis deals with the ideas of calculus in a rigorous, axiomatic way. Stein wrote four books in the series. The first was on Fourier analysis (the art and science of decomposing arbitrary signals into combinations of simple harmonic waves). The next was on complex analysis (which treats functions that have complex numbers as both inputs and outputs), followed by real analysis (which develops, among other things, a rigorous way to measure sizes of sets) and finally functional analysis (which deals broadly with functions of functions). These are core subjects containing foundational knowledge for any working mathematician.

In Stein’s class, my fellow students and I were the guinea pigs on whom the material for his books was to be rehearsed. We had front-row seats as Eli (as I later came to call him) showcased his beloved subject’s far-reaching consequences: Look at how amazing analysis is, he would say. You can even use it to resolve problems in the distant world of number theory! Indeed, his book on Fourier analysis builds up to a proof of Dirichlet’s theorem on primes in arithmetic progressions, which says, for example, that infinitely many primes leave a remainder of 6 when divided by 35 (since 6 and 35 have no prime factors in common). And his course on complex analysis included a proof of the prime number theorem, which gives an asymptotic estimate for the number of primes below a growing bound. Moreover, I learned that if the Riemann hypothesis is true, we’ll get a much stronger prime number theorem than the one known today. To see why that is and for a closer look under the hood of this famous math problem, please watch the accompanying video at the top of this page.

Despite Eli’s proselytizing to us on the wide-ranging power of analysis, I learned the opposite lesson: Look at how amazing number theory is — you can even use fields as far away as analysis to prove the things you want! Stein’s class helped set me on the path to becoming a number theorist. But as I came to understand more about the Riemann hypothesis over the years, I learned not to make it a focus of my research. It’s just too hard to make progress.

After Princeton, I went off to graduate school at Columbia University. It was an exciting time to be working in number theory. In 2003, Dan Goldston and Cem Yıldırım announced a spectacular new result about gaps in the primes, only to withdraw the claim soon after. (As Goldston wrote years later on accepting the prestigious Cole Prize for these ideas: “While mathematicians often do not have much humility, we all have lots of experience with humiliation.”) Nevertheless, the ideas became an important ingredient in the Green-Tao theorem, which shows that the set of prime numbers contains arithmetic progressions of any given length. Then, working with János Pintz, Goldston and Yıldırım salvaged enough of their method to prove, in their breakthrough GPY theorem in 2005, that primes will infinitely often have gaps which are arbitrarily small when compared to the average gap. Moreover, if you could improve their result by any amount at all, you would prove that primes infinitely often differ by some bounded constant. And this would be a huge leap toward solving the notoriously difficult twin primes conjecture , which predicts that there are infinitely many pairs of primes that differ by 2.

A meeting on how to extend the GPY method was immediately organized at the American Institute of Mathematics in San Jose, California. As a bright-eyed and bushy-tailed grad student, I felt extraordinarily lucky to be there among the world’s top experts. By the end of the week, the experts agreed that it was basically impossible to improve the GPY method to get bounded prime gaps. Fortunately, Yitang Zhang did not attend this meeting. Almost a decade later, after years of incredibly hard work in relative isolation, he found a way around the impasse and proved the experts wrong . I guess the moral of my story is that when people organize meetings on how not to solve the Riemann hypothesis (as they do from time to time), don’t go!

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Riemann hypothesis

In 2001, the University of Texas, Austin held a series of seven general audience evening lectures, “The Millennium Lectures”, based on the “Millennium Prize Problems.” Their aim was to explain to a wide audience the historical background to these problems, why they have resisted many years of serious attempts to solve them, and the roles these problems play in modern mathematics. The lecture series ran throughout the Spring Semester 2001 and were given by members of Departments of Mathematics and Computer Science.

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The Riemann Hypothesis, the Biggest Problem in Mathematics, Is a Step Closer to Being Solved

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