riemann hypothesis proof 2022

A new approach to a $1 million mathematical enigma

Physicist translates the riemann zeta function into quantum field theory.

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

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

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

An outside perspective

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

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

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

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

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

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

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

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

Deceptively simple

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

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

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

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

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

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

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

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

A physical analogue

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

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

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

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

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

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

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

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• 11 November 2022
• Correction 14 November 2022

Mathematician who solved prime-number riddle claims new breakthrough

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

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Nature 611 , 645-646 (2022)

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

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

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

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

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

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Proof of the Riemann Hypothesis

𝑠 1 d 𝑥 \zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\frac{x-\lfloor x\rfloor}{x^{s+1}}\,\text{d}x and solving the integral for the real- and imaginary part of the zeta function.

1 Introduction

In 1859 Bernhard Riemann found one of the most eminent mathematical problems of our time: In his paper “On the Number of Primes Less Than a Given Magnitude” [4] he published the assumption that all non-trivial zero-points of the zeta function extended to the range of complex numbers ℂ ℂ \mathbb{C} have a real part of 1 2 1 2 \frac{1}{2} , noting the demand of a strict proof for this. Ever since David Hilbert in 1900 added this problem to his list of the 23 most important problems of 20 th th {}^{\text{th}} century, mathematicians have been working on finding evidence for Riemanns hypothesis. This paper aims to provide the proof and fill this gap in modern mathematics.

2 Proof of the Riemann Hypothesis

The zeta-function ζ ​ ( s ) 𝜁 𝑠 \zeta(s) in the complex range s ∈ ℂ 𝑠 ℂ s\in\mathbb{C} for a positive real-part of s 𝑠 s can be formulated as integral representation

𝑎 𝑖 𝑏 𝑎 𝑏 ℝ s=a+ib;\,a,b\in\mathbb{R} and 0 < a < 1 0 𝑎 1 0<a<1 as well as 0 < b 0 𝑏 0<b . Be s 0 subscript 𝑠 0 s_{0} a zero point of the zeta function. From [1] we know, that the zeta-function is symmetrical in a way that ζ ​ ( s 0 ) = 0 ⇔ ζ ​ ( 1 − s 0 ) = 0 ⇔ 𝜁 subscript 𝑠 0 0 𝜁 1 subscript 𝑠 0 0 \zeta(s_{0})=0\Leftrightarrow\zeta(1-s_{0})=0 for all zero-points s 0 ∈ ℂ subscript 𝑠 0 ℂ s_{0}\in\mathbb{C} (see appendix for details). In accordance to equation 1 we can write ζ ​ ( 1 − s ) 𝜁 1 𝑠 \zeta(1-s) as

𝑎 𝑖 𝑏 s_{0}=a+ib into ζ ​ ( s ) 𝜁 𝑠 \zeta(s) , using x − 1 − a − i ​ b = x − 1 − a ​ cos ⁡ ( b ​ ln ⁡ ( x ) ) − i ​ x − 1 − a ​ sin ⁡ ( b ​ ln ⁡ ( x ) ) superscript 𝑥 1 𝑎 𝑖 𝑏 superscript 𝑥 1 𝑎 𝑏 𝑥 𝑖 superscript 𝑥 1 𝑎 𝑏 𝑥 x^{-1-a-ib}=x^{-1-a}\cos\left(b\ln(x)\right)-ix^{-1-a}\sin\left(b\ln(x)\right) and defining { x } := x − ⌊ x ⌋ assign 𝑥 𝑥 𝑥 \{x\}:=x-\lfloor x\rfloor we get the following two equations 3 and 4 out off ζ ​ ( s ) 𝜁 𝑠 \zeta(s) and ζ ​ ( 1 − s ) 𝜁 1 𝑠 \zeta(1-s) :

Thus, we get 4 equations, for the real- and imaginary-part by means of ζ ​ ( s 0 ) 𝜁 subscript 𝑠 0 \zeta(s_{0}) (being called here ℜ 1 subscript 1 \Re_{1} and ℑ 1 subscript 1 \Im_{1} ) and ζ ​ ( 1 − s 0 ) 𝜁 1 subscript 𝑠 0 \zeta(1-s_{0}) (being called here ℜ 2 subscript 2 \Re_{2} and ℑ 2 subscript 2 \Im_{2} ):

In the following we want to solve the integrals of ℜ 1 subscript 1 \Re_{1} , ℜ 2 subscript 2 \Re_{2} , ℑ 1 subscript 1 \Im_{1} and ℑ 2 subscript 2 \Im_{2} via partial integration. As commonly known, it is

1 𝑎 𝑏 𝑥 v^{\prime}(x):=\frac{1}{x^{1+a}}\cos(b\ln(x)) , thus

Using equation 7 we can calculate ∫ u ′ ​ ( x ) ⋅ v ​ ( x ) ​ d ​ x = ∫ 1 ⋅ v ​ ( x ) ​ d ​ x ⋅ superscript 𝑢 ′ 𝑥 𝑣 𝑥 d 𝑥 ⋅ 1 𝑣 𝑥 d 𝑥 \int u^{\prime}(x)\cdot v(x)\,\text{d}x=\int 1\cdot v(x)\,\text{d}x as

Thus, using { 1 } = 0 1 0 \{1\}=0 and ln ⁡ ( 1 ) = 0 1 0 \ln(1)=0 , we can write ℜ 1 subscript 1 \Re_{1} as

For solving the integral of ℜ 2 subscript 2 \Re_{2} we define v ′ ​ ( x ) := 1 x 2 − a ​ cos ⁡ ( b ​ ln ⁡ ( x ) ) assign superscript 𝑣 ′ 𝑥 1 superscript 𝑥 2 𝑎 𝑏 𝑥 v^{\prime}(x):=\frac{1}{x^{2-a}}\cos(b\ln(x)) , therefore we have

Using equation 10 we can calculate ∫ u ′ ​ ( x ) ⋅ v ​ ( x ) ​ d ​ x = ∫ 1 ⋅ v ​ ( x ) ​ d ​ x ⋅ superscript 𝑢 ′ 𝑥 𝑣 𝑥 d 𝑥 ⋅ 1 𝑣 𝑥 d 𝑥 \int u^{\prime}(x)\cdot v(x)\,\text{d}x=\int 1\cdot v(x)\,\text{d}x as

In accordance to above, we can write ℜ 2 subscript 2 \Re_{2} as

1 𝑎 𝑏 𝑥 v^{\prime}(x):=\frac{1}{x^{1+a}}\sin(b\ln(x)) , therefore we have

Using equation 13 we can calculate ∫ u ′ ​ ( x ) ⋅ v ​ ( x ) ​ d ​ x = ∫ 1 ⋅ v ​ ( x ) ​ d ​ x ⋅ superscript 𝑢 ′ 𝑥 𝑣 𝑥 d 𝑥 ⋅ 1 𝑣 𝑥 d 𝑥 \int u^{\prime}(x)\cdot v(x)\,\text{d}x=\int 1\cdot v(x)\,\text{d}x as

In accordance to above, we can write ℑ 1 subscript 1 \Im_{1} as

In the same way for solving the integral of ℑ 2 subscript 2 \Im_{2} we define v ′ ​ ( x ) := 1 x 2 − a ​ sin ⁡ ( b ​ ln ⁡ ( x ) ) assign superscript 𝑣 ′ 𝑥 1 superscript 𝑥 2 𝑎 𝑏 𝑥 v^{\prime}(x):=\frac{1}{x^{2-a}}\sin(b\ln(x)) , therefore we have

Using equation 16 we can calculate ∫ u ′ ​ ( x ) ⋅ v ​ ( x ) ​ d ​ x = ∫ 1 ⋅ v ​ ( x ) ​ d ​ x ⋅ superscript 𝑢 ′ 𝑥 𝑣 𝑥 d 𝑥 ⋅ 1 𝑣 𝑥 d 𝑥 \int u^{\prime}(x)\cdot v(x)\,\text{d}x=\int 1\cdot v(x)\,\text{d}x as

In accordance to above, we can write ℑ 2 subscript 2 \Im_{2} as

superscript 𝑎 1 2 superscript 𝑏 2 (a^{2}+b^{2})((a-1)^{2}+b^{2}) we can simplify to:

With the equations 20 , it follows from ℜ 2 subscript 2 \mathbb{\Re}_{2} :

Accordingly from ℑ 2 subscript 2 \Im_{2} we can extract

Equating Eq. 21 and Eq. 22 provides

In the same way reshaping ℜ 1 subscript 1 \Re_{1} like

as well as ℑ 1 subscript 1 \Im_{1} like

with subsequent equating of Eq. 24 and 25 we acquire

superscript 𝑎 1 2 superscript 𝑏 2 0 a((a-1)^{2}+b^{2})\neq 0 . Thus, a = 1 2 𝑎 1 2 a=\frac{1}{2} follows from Eq. 27 = Eq. 28 . Ergo, a = 1 2 𝑎 1 2 a=\frac{1}{2} is the only valid solution for the Zeta function to become zero in the critical stripe and the Riemann Hypothesis is true, Q.E.D..

3 Conclusion

In this paper we have proven the Riemann hypothesis, stating that the real part of all zero points of the zeta function is 1 2 1 2 \frac{1}{2} , to be true. For this we have used the integral-representation of ζ ​ ( s ) 𝜁 𝑠 \zeta(s) , solved the integral and thereby formulated conditions for ζ ​ ( s ) 𝜁 𝑠 \zeta(s) to be zero. Utilizing those conditions, we could show, that s 𝑠 s must have a real part of 1 2 1 2 \frac{1}{2} for ζ ​ ( s ) 𝜁 𝑠 \zeta(s) to be zero.

According to Gelbart et. al. (see [1]) ζ ​ ( s ) 𝜁 𝑠 \zeta(s) and ζ ​ ( 1 − s ) 𝜁 1 𝑠 \zeta(1-s) are connected by a function G ​ ( s ) 𝐺 𝑠 G(s) via

Since π x ≠ 0 superscript 𝜋 𝑥 0 \pi^{x}\neq 0 and Γ ​ ( x ) ≠ 0 ​ ∀ x ∈ ℂ Γ 𝑥 0 for-all 𝑥 ℂ \Gamma(x)\neq 0\ \forall\,x\in\mathbb{C} determines G ​ ( s ) ≠ 0 ​ ∀ s ∈ ℂ 𝐺 𝑠 0 for-all 𝑠 ℂ G(s)\neq 0\ \forall\,s\in\mathbb{C} . Be s 0 subscript 𝑠 0 s_{0} a zero-point of the zeta-function. Because of G ​ ( s 0 ) ≠ 0 𝐺 subscript 𝑠 0 0 G(s_{0})\neq 0 it is

S. Gelbart and S. Miller. Riemann’s zeta function and beyond. Bulletin of the American Mathematical Society , 41:59 112, 2003

Yuri Heymann. An investigation of the non-trivial zeros of the riemann zeta function. arXiv ,1804.04700, 2020.

Bronstein, Semendjajew, Musiol, Muehlig. Taschenbuch der mathematik. 2008.

B. Riemann. Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse. Monatsberichte der Berliner Akademie , 1859.

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Science News

Here’s why we care about attempts to prove the riemann hypothesis.

The latest effort shines a spotlight on an enduring prime numbers mystery

LINED UP   The Riemann zeta function has an infinite number of points where the function’s value is zero, located at the whirls of color in this plot. The Riemann hypothesis predicts that certain zeros lie along a single line, which is horizontal in this image, where the colorful bands meet the red.

Empetrisor/Wikimedia Commons ( CC BY-SA 4.0 )

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By Emily Conover

September 25, 2018 at 11:46 am

A famed mathematical enigma is once again in the spotlight.

The Riemann hypothesis, posited in 1859 by German mathematician Bernhard Riemann, is one of the biggest unsolved puzzles in mathematics. The hypothesis, which could unlock the mysteries of prime numbers, has never been proved. But mathematicians are buzzing about a new attempt.

Esteemed mathematician Michael Atiyah took a crack at proving the hypothesis in a lecture at the Heidelberg Laureate Forum in Germany on September 24. Despite the stature of Atiyah — who has won the two most prestigious honors in mathematics, the Fields Medal and the Abel Prize — many researchers have expressed skepticism about the proof. So the Riemann hypothesis remains up for grabs.

Let’s break down what the Riemann hypothesis is, and what a confirmed proof — if one is ever found — would mean for mathematics.

What is the Riemann hypothesis?

The Riemann hypothesis is a statement about a mathematical curiosity known as the Riemann zeta function. That function is closely entwined with prime numbers — whole numbers that are evenly divisible only by 1 and themselves. Prime numbers are mysterious: They are scattered in an inscrutable pattern across the number line, making it difficult to predict where each prime number will fall ( SN Online: 4/2/08 ).

But if the Riemann zeta function meets a certain condition, Riemann realized, it would reveal secrets of the prime numbers, such as how many primes exist below a given number. That required condition is the Riemann hypothesis. It conjectures that certain zeros of the function — the points where the function’s value equals zero — all lie along a particular line when plotted ( SN: 9/27/08, p. 14 ). If the hypothesis is confirmed, it could help expose a method to the primes’ madness.

Why is it so important?

Prime numbers are mathematical VIPs: Like atoms of the periodic table, they are the building blocks for larger numbers. Primes matter for practical purposes, too, as they are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.”

Haven’t people tried to prove this before?

Yep. It’s difficult to count the number of attempts, but probably hundreds of researchers have tried their hands at a proof. So far none of the proofs have stood up to scrutiny. The problem is so stubborn that it now has a bounty on its head : The Clay Mathematics Institute has offered up $1 million to anyone who can prove the Riemann hypothesis.

Why is it so difficult to prove?

The Riemann zeta function is a difficult beast to work with. Even defining it is a challenge, Ono says. Furthermore, the function has an infinite number of zeros. If any one of those zeros is not on its expected line, the Riemann hypothesis is wrong. And since there are infinite zeros, manually checking each one won’t work. Instead, a proof must show without a doubt that no zero can be an outlier. For difficult mathematical quandaries like the Riemann hypothesis, the bar for acceptance of a proof is extremely high. Verification of such a proof typically requires months or even years of double-checking by other mathematicians before either everyone is convinced, or the proof is deemed flawed.

What will it take to prove the Riemann hypothesis?

Various mathematicians have made some amount of headway toward a proof. Ono likens it to attempting to climb Mount Everest and making it to base camp. While some clever mathematician may eventually be able to finish that climb, Ono says, “there is this belief that the ultimate proof … if one ever is made, will require a different level of mathematics.”

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A professor’s work on prime numbers could solve a 150-year-old puzzle in math

His work will be published soon..

Ameya Paleja

Yitang Zhang.

Wikimedia Commons  

Shanghai-born Zhang Yitang is a professor of mathematics at the University of California, Santa Barbara. If a 111-page manuscript allegedly written by him passes peer review, he might become the first person to solve the Riemann hypothesis, The South China Morning Post (SCMP) has reported.

The Riemann hypothesis is a 150-year-old puzzle that is considered by the community to be the holy grail of mathematics. Published in 1859, it is a fascinating piece of mathematical conjecture around prime numbers and how they can be predicted.

Riemann hypothesized that prime numbers do not occur erratically but rather follow the frequency of an elaborate function, which is called the Riemann zeta function. Using this function, one can reliably predict where prime numbers occur, but more than a century later, no mathematician has been able to prove this hypothesis.

Who is Zhang Yitang?

Born in 1955, Zhang could not attend school and taught himself mathematics at the age of 11. He worked in the fields and factories for several years to make his way to Peking University, where he earned his master’s degree in 1984.

Zhang then moved to the U.S. to get a Ph.D. in mathematics from Purdue University. Failing to get himself a job, Zhang then worked as an accountant, a restaurant manager, and even a food delivery person before getting a position to teach pre-algebra and calculus at the University of New Hampshire in 1999, the SCMP report said.

In 2013, Zhang shocked the world with his twin prime conjecture, which proposed that there were an infinite pair of prime numbers that differed by two. Prior to this, Zhang had achieved only one publication.

What has Zhang done now?

A manuscript that is allegedly written by Zhang has now surfaced in the mathematics research community and has proof related to the Riemann hypothesis. Although the paper has not been peer-reviewed or verified by Zhang himself, if found accurate by the mathematical community, it would mean the end of another famous mathematical hypothesis, the Landau-Siegel conjecture.

Named after mathematicians Edmund Landau and Carl Siegel, the conjecture speaks about the existence of zero points of type of L-functions in number theory. Simply put, the conjecture provides counterexamples to the Riemann hypothesis.

Zhang is expected to present his work at a lecture at Peking University today, and the publication could possibly enter the peer review process this month, the SCMP report said. The outcome of the process will be known in a few months’ time, and if found accurate, could land Zhang a $1 million prize from the Clay Mathematics Institute.

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This is not the first instance of a claim made for the Clay Institute’s prize. Last year, media reports suggested that a mathematics professor in India had submitted such proof, while another famous mathematician Sir Michael Atiyah made similar claims in 2018. The Clay Institute has rejected both claims and confirmed that the Riemann hypothesis remains unsolved .

Will it be different this time around? The paper must pass peer review first.

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Ameya Paleja Ameya&nbsp;is a science writer based in Hyderabad, India. A Molecular Biologist at heart, he traded the micropipette to write about science during the pandemic and does not want to go back. He likes to write about genetics, microbes, technology, and public policy.

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A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

How to cite: Zhang, W. A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$. Preprints 2021 , 2021080146. https://doi.org/10.20944/preprints202108.0146.v21 Zhang, W. A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$. Preprints 2021, 2021080146. https://doi.org/10.20944/preprints202108.0146.v21 Copy

Zhang, W. A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$. Preprints 2021 , 2021080146. https://doi.org/10.20944/preprints202108.0146.v21

Zhang, W. (2022). A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$. Preprints. https://doi.org/10.20944/preprints202108.0146.v21

Zhang, W. 2022 "A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$" Preprints. https://doi.org/10.20944/preprints202108.0146.v21

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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