riemann hypothesis solved 2022

November 15, 2022

Mathematician Who Solved Prime Number Riddle Claims New Breakthrough

After shocking the mathematics community with a major result in 2013, Yitang Zhang now says he has solved an analogue of the celebrated Riemann hypothesis

By Davide Castelvecchi & Nature magazine

Prime numbers.

Science Photo Library/Alamy Stock Photo

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.

Number theorist Yitang Zhang, who is based at the University of California, Santa Barbara, posted his proposed solution—a 111-page preprint—on the arXiv preprint server on 4 November. It has not yet been validated by his peers. But if it checks out, it will go some way towards taming the randomness of prime numbers, whole numbers that cannot be divided evenly by any number except themselves or 1.

The Landau–Siegel zeros conjecture is similar to—and, some suspect, less challenging than—the Riemann hypothesis, another question on the randomness of primes and one of the biggest unsolved mysteries in mathematics. Although it has been known for millennia that there are infinitely many prime numbers, there is no way to predict whether a given number will be prime; only the probability that it will be, given its size. Solving either the Riemann or Landau-Siegel problems would mean that the distribution of prime numbers does not have huge statistical fluctuations.

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“For me in the field, this result would be massive,” says Andrew Granville, a number theorist at the University of Montreal in Canada. But he warns that others, including Zhang, have previously proposed solutions that turned out to be faulty, and that it will take a while for researchers to comb through Zhang’s argument to see if it is correct. “Right now, we’re very far from being certain.”

Zhang did not respond to Nature ’s requests for comment. But he did write about his latest work on the Chinese website Zhihu . “As for the Landau–Siegel zeros conjecture, I didn’t think about giving up,” he wrote. He added: “As for my planning of the future, I won’t give away these math problems. I think I probably have to do mathematics all my life. I don’t know what to do without doing mathematics. People have asked questions about my retirement. I’ve said that if I leave math, I really won’t know how to live.” (His comments were translated into English by the website Pandaily .)

Passion for primes

Rumours had been circulating since mid-October that Zhang had made a breakthrough on the Landau–Siegel problem, and the mathematics community is certain to pay attention. Zhang has only one significant result to his name, but it is one for the ages. For years after attaining his PhD in 1991, he was estranged from his thesis adviser, working odd jobs to make ends meet. He then took up a teaching position at the University of New Hampshire in Durham, where he quietly chiselled away at his passion, the statistical properties of prime numbers. He posted a preprint on the Landau–Siegel conjecture in 2007, but mathematicians found problems and it was never published in a peer-reviewed journal.

Zhang’s first big breakthrough came in 2013, when he showed that although the gaps between subsequent primes grow larger and larger on average, there are infinitely many pairs that stay within a certain finite distance of each other. This was the first big step towards solving a major question in number theory—whether there are infinitely many pairs of primes that differ by just 2 units, such as the primes 5 and 7 or 11 and 13. (Number theorist James Maynard at the University of Oxford, UK, won a Fields Medal in July for improving on Zhang’s result, among other achievements.)

The problem Zhang now says he has solved dates back to the turn of the twentieth century, when mathematicians were exploring ways to tame the randomness of prime numbers. One way to count them is to partition them into a finite number of baskets, based on the remainders one gets when dividing a prime by another prime, denoted by p . For example, when divided by p = 5, a prime can give a remainder of 1, 2, 3 or 4. A result from the early nineteenth century shows that—once one considers a large enough statistical sample—these possibilities should ‘eventually’ occur with equal probability. But the big question, Granville explains, was how large the statistical sample should be for the equal-distribution pattern to show up: “What does ‘eventually’ mean? When do they start becoming well distributed?”

The methods known at the time suggested that the samples should be stupendously large, growing exponentially with the size of p . But a German mathematician called Carl Ludwig Siegel found a relatively simple formula that linked to this basket problem, and potentially made the samples much smaller. He showed that if, under certain circumstances, the formula did not yield 0, this was tantamount to proving the conjecture. “He removed all the dead wood out of the way and left just one massive oak to be felled,” Granville says. The problem, also formulated independently by another German mathematician, Edmund Landau, became known as the Landau–Siegel zeros conjecture. What Zhang now claims to have proved is a weaker version of it, but one that would have similar consequences regarding the distribution of primes.

Unsolved problem

The conjecture is a cousin of the Riemann hypothesis—a way to predict the probability that numbers in a certain range are prime that was devised by German mathematician Bernhard Riemann in 1859.

The Riemann hypothesis will probably remain at the top of mathematicians’ wishlists for years to come. Despite its importance, no attempts so far have made much progress. Only the bravest of mathematicians—often those who already have major accomplishments and prizes under their belts—publicly admit to trying to solve it. “It’s one of those things—you’re not supposed to talk about Riemann,” says Alex Kontorovich, a number theorist at Rutgers University in Piscataway, New Jersey. “People work secretly on it.”

Although progress towards solving the Riemann hypothesis has stalled, the Landau–Siegel problem offers similar insights, he adds. “Resolving any of these issues would be a major advancement in our understanding of the distribution of prime numbers.”

This article is reproduced with permission and was first published on November 11 2022.

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 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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Mathematicians Clear Hurdle in Quest to Decode Primes

January 13, 2022

Mia Carnevale for Quanta Magazine

Introduction

It’s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. Despite their best efforts, mathematicians have made very little progress on the Riemann hypothesis . But they have managed to make headway on simpler related problems.

In a paper posted in September , Paul Nelson of the Institute for Advanced Study has solved a version of the subconvexity problem, a kind of lighter-weight version of Riemann’s question. The proof is a significant achievement on its own and teases the possibility that even greater discoveries related to prime numbers may be in store.

“It’s a bit of a far-fetched dream, but you could hyper-optimistically hope that maybe we get some insight in how the [Riemann hypothesis] works by working on problems like this,” Nelson said.

The Riemann hypothesis and the subconvexity problem are important because prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics. When you plot them on the number line, there appears to be no pattern to how they’re distributed. But in 1859 Riemann devised an object called the Riemann zeta function — a kind of infinite sum — which fueled a revolutionary approach that, if proved to work, would unlock the primes’ hidden structure.

“It proves a result that a few years ago would have been regarded as science fiction,” said Valentin Blomer of the University of Bonn.

Getting Complex

Riemann’s question hinges on the Riemann zeta function. The terms it adds together are the reciprocals of the whole numbers, in which the denominators are raised to a power defined by a variable, s (so $latex \frac{1}{1^{s}}$, $latex \frac{1}{2^{s}}$, $latex \frac{1}{3^{s}}$ and so on).

Riemann proposed that if mathematicians could prove a basic property of this function — what it takes for it to equal zero — they’d be able to estimate with great accuracy how many prime numbers there are along any given interval on the number line.

Prior to Riemann, Leonhard Euler constructed a similar function and used it to create a new proof that there are infinitely many primes. In Euler’s function, the denominators are raised to powers that are real numbers. The Riemann zeta function, by contrast, assigns complex numbers to the variable s , an innovation that brings the whole vast store of techniques from complex analysis to bear on questions in number theory.

Complex numbers have two parts, one real and one imaginary, the latter of which relates to the imaginary number i , defined as the square root of −1. Examples include 3 + 4 i and 2 − 6 i . In these cases, the 3 and the 2 are the real parts, while the 4 and −6 are the imaginary parts.

The Riemann hypothesis is about which values of s make the Riemann zeta function equal zero. It predicts that the only important, or nontrivial, values of s that do this are complex numbers whose real part equals $latex \frac{1}{2}$. (The function also equals zero whenever s is a negative even integer with an imaginary part that equals zero, but those zeros are easy to see and are considered trivial.) If the Riemann hypothesis is true, the Riemann zeta function explains how primes are distributed on the number line. (Exactly how it explains that is complicated. Quanta recently produced a video detailing just how it works .)

In the years since Riemann proposed it, the Riemann hypothesis has instigated many advances in mathematics, though mathematicians have made little progress on the question itself. Given that relative futility, they’ve at times redirected their attention to slightly easier questions which approximate Riemann’s intractable riddle.

Next to Nothing

The problem Paul Nelson solved is two steps removed from the Riemann hypothesis. Each step takes a bit of explanation.

The first is the Lindelöf hypothesis. Where the Riemann hypothesis says that the only nontrivial zeroes of the Riemann zeta function occur when the real part of s equals $latex \frac{1}{2}$, the Lindelöf hypothesis merely says that under that condition, the output of the Riemann zeta function is small in a certain precise sense.

For both the Riemann and Lindelöf hypotheses, the real part of s is fixed at $latex \frac{1}{2}$, but the imaginary part can be any number you like: 2, 537, $latex \frac{1}{2}$ . One way to define “small” is to compare the number of digits in the input, s , with the number of digits in the output.

Mathematicians can easily establish that the output never has more than 25% as many digits as the input. This means that it grows as the input grows, but it doesn’t grow disproportionately. This 25% is called the trivial bound. But the Lindelöf hypothesis says that as the inputs get larger, the size of the output is actually always bounded at 1% as many digits as the input.

For more than a century, mathematicians have worked on closing the gap between the trivial bound (25%) and the conjectured bound (1%). They have made a dozen or so improvements, the most recent in 2017 when Jean Bourgain proved that for values of s with real part $latex \frac{1}{2}$, the output of the Riemann zeta function has a size that is about 15% the size of the input. So if the input is a 1,000,000-digit number, the output won’t have more than 150,000 digits. It’s a far cry from proving the Lindelöf hypothesis, let alone Riemann’s question, but it’s something.

“We haven’t made any progress on the Riemann hypothesis in 150 years, whereas this is a question we can make incremental progress towards,” said Nelson. “There’s a way you can kind of keep score.”

The Lindelöf hypothesis is just one example of a Riemann-adjacent problem amenable to scorekeeping. In his new work, Nelson solved another problem that’s one more step removed from Riemann’s question.

Families of Functions

The Riemann zeta function is the most famous member of a large class of mathematical objects, L -functions, that encode many different arithmetic relationships. By modifying the definition of the Riemann zeta function, mathematicians construct other L -functions that provide more refined information about the primes. For example, the properties of some L -functions measure how many primes below a certain value have a given number as their last digit.

Due to this versatility, L -functions are objects of intense study, and they are central players in a sprawling research vision known as the Langlands program . For now, mathematicians still lack a full theory explaining just what they are.

“There is some big zoo of these things, and for most of them we can’t prove anything at all,” said Nelson.

Video : Alex Kontorovich, professor of mathematics at Rutgers University, breaks down the notoriously difficult Riemann hypothesis in this comprehensive explainer.

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

One piece of that theory involves a generalization of the Lindelöf hypothesis, which predicts that whenever the real part of the complex number input equals $latex \frac{1}{2}$, the output stays small relative to the input for all L -functions (not just the Riemann zeta function).

While mathematicians have chipped away at the Lindelöf hypothesis, they’d only managed scattered progress on something known as the subconvexity problem. Solving that simply amounts to breaking the trivial bound — that is, proving that for any L -function, the output will have less than 25% of the number of digits of the input (multiplied by a quantity called the degree of the L -function). Previously, mathematicians managed to do that for only a few specific families of L -functions (including the Riemann zeta function) and were far from achieving a general result.

But that began to change in the 1990s when mathematicians recognized that merely breaking the trivial bound for general L -functions could lead to advances on different problems, including  questions in an area of research called arithmetic quantum chaos and a question about which integers can be written as sums of three squares.

“People realized in the last 20 to 30 years that there are all these problems that could be solved, provided one could prove this technical-looking statement” about subconvexity, said Emmanuel Kowalski of the Swiss Federal Institute of Technology Zurich.

Nelson was the mathematician who finally did it, after two decades of work that helped him learn how to imagine it.

A Shift in Perspective

In the early 2000s two teams of mathematicians — Joseph Bernstein and Andre Reznikov on one team, and Philippe Michel and Akshay Venkatesh on the other — transformed how mathematicians estimate L -functions. Instead of seeing them merely in arithmetic terms, they created a geometric way to think about the size of their outputs. That work contributed to Venkatesh winning the Fields Medal , math’s highest honor, in 2018.

In this revised picture, the size of an L -function is linked to the size of an integral, called a period, that can be calculated by integrating a function called an automorphic form along a geometric space. This provided mathematicians with more tools they could use to try and break the trivial bound.

“You had more techniques to play with,” said Michel, of the Swiss Federal Institute of Technology Lausanne.

Nelson and Venkatesh collaborated on a 2018 paper that determined which automorphic forms are best for making the kinds of size estimates needed to answer the subconvexity problem. In the following years, Nelson produced two more solo papers on the topic — the first in 2020, the second this past September — that together solved it.

Nelson proved that each L -function satisfies a subconvex bound, meaning its outputs are less than 25% the size of its inputs. He broke the bound by a hair — getting just below 25% for most L -functions — but sometimes that’s all it takes to cross from one world into the next.

“He broke trivial bound, and we are amply satisfied with this. It’s really the breaking of things,” said Michel.

Now mathematicians will march their subconvex bound off to face other problems, maybe even including the Riemann hypothesis one day. That may seem far-fetched right now, but math thrives on hope, and at the very least, Nelson’s new proof has provided that.

Correction : January 13, 2022 A previous version of this article stated that the Riemann hypothesis predicts that the only non-trivial zeroes of the Riemann zeta function occur whenever the real part of  s  is $latex \frac{1}{2}$. It actually predicts nontrivial zeroes occur only when the real part equals $latex \frac{1}{2}$. The article has been updated accordingly.

Correction : January 14, 2022 This article has been updated to emphasize that Nelson solved a version of the subconvexity problem, not the full problem.

Correction : January 18, 2022 A previous version of this article incorrectly included “ i ”  when discussing the imaginary part of a complex number. The article has been updated accordingly.

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Mathematician who solved prime-number riddle claims new breakthrough

"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 .

Number theorist Yitang Zhang , who is based at the University of California, Santa Barbara, posted his proposed solution—a 111-page preprint—on the arXiv preprint server on 4 November 1 . It has not yet been validated by his peers. But if it checks out, it will go some way towards taming the randomness of prime numbers, whole numbers that cannot be divided evenly by any number except themselves or 1."

Read more at  Nature .

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Retired mathematician rocks math world with claim that he's solved $1 million problem

A famous problem in mathematics that has remained unsolved for almost 160 years probably still is — despite a new claim by a retired mathematician that he has cracked it .

Michael Atiyah, a professor emeritus at the University of Edinburgh in Scotland, announced on Monday at a scientific conference in Heidelberg, Germany , that he had devised a "simple proof" for the Riemann Hypothesis, a problem relating to patterns of prime numbers that has defied efforts to solve it since it was first proposed in 1859.

If Atiyah is right about his proof, he could claim a $1 million prize set aside in 2000 for the first person to prove the hypothesis. A proof of Riemann, which would provide a sort of "map" of prime numbers, could also have implications for cryptography and other fields beyond mathematics.

But Atiyah's peers are dubious of the claim.

"Atiyah is a wizard of a mathematician, but there's a lot of skepticism among mathematicians that his wizardry has been sufficient to crack the Riemann Hypothesis," John Allen Paulos, a professor of mathematics at Temple University in Philadelphia and the author of several popular books on mathematical topics, told NBC News MACH in an email.

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John Baez, a mathematical physicist at the University of California, Riverside, offered an even blunter assessment. "I know of nobody who believes Atiyah has actually proved the Riemann Hypothesis," he said in an email. "I've looked at his papers on this. His arguments just stack one impressive claim on top of another, without any logical argument."

But Atiyah, who has won many prestigious prizes in mathematics over a career spanning more than 60 years, is undeterred. "All radical ideas meet with initial skepticism," he said in an email, adding that he would soon publish an expanded version of the proof that is now available online in an abbreviated form .

The Riemann Hypothesis was first proposed by German mathematician Bernhard Riemann in a six-page paper published in 1859. It posits that there is a definite pattern to the distribution of prime numbers like 2, 3, 5, 7, 11, 13 and so on (numbers that can't be expressed as the product of two smaller numbers), which occur less frequently as one moves up the number line. The hypothesis has been checked for the first 10 trillion solutions, but no one has proven that it's true for all prime numbers.

Whether or not Atiyah has proved Riemann could take some time to figure out. Even if he delivers the full proof as promised, it will have to undergo years of scrutiny before it's accepted by the Clay Mathematics Institute , the Peterborough, New Hampshire-based foundation that is offering the prize.

Mathematical proofs often turn out to be incorrect. In one well-known example, British mathematician Andrew Wiles in 1993 offered a proof of another famous problem, Fermat's Last Theorem , that turned out to contain a mistake. But he was able to correct the error, and he published the corrected version in 1995.

"The only unusual thing about [Atiyah's purported proof] is the high-profile way in which he's chosen to share it," Katie Steckles, a mathematician in Manchester, England, said in an email. "Normally this kind of work would be shared with colleagues before a formal public presentation."

But Steckles defended Atiyah, calling him a "very well-respected mathematician with a long history of great work. ... While it's great that he's still working on mathematics up to his retirement, it would be a shame if people were harsh in criticism of him at this stage."

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Department of Mathematics

Mathematicians clear hurdle in quest to decode primes.

D-MATH in the Media

Paul Nelson has solved the subconvexity problem, bringing mathematicians one step closer to understanding the Riemann hypothesis and the distribution of prime numbers.

• mode_comment Number of comments

It’s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. Despite their best efforts, mathematicians have made very little progress on the Riemann hypothesis. But they have managed to make headway on simpler related problems. In a paper posted in September, Paul Nelson of the Institute for Advanced Study has solved the subconvexity problem, a kind of lighter-weight version of Riemann’s question. The proof is a significant achievement on its own and teases the possibility that even greater discoveries related to prime numbers may be in store.

Read full article in external page Quanta Magazine call_made

Paul Nelson was an assistant professor at the Department from August 2014 to July 2021. Since September 2021, he has been von Neumann Fellow at the Institute for Advanced Study in Princeton.

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• 14 November 2022

Daily briefing: Mathematician claims a prime-number problem breakthrough

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Number theorist Yitang Zhang tackled a problem that could tame the randomness of prime numbers. Credit: George Csicsery/Zala Films

Mathematician claims new breakthrough

Number theorist Yitang Zhang, 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. Zhang posted his proposed solution — a 111-page preprint — on arXiv, and it has not yet been validated by his peers. If it checks out, it will help to tame the randomness of prime numbers, but Zhang and other scientists have previously proposed solutions to this problem that turned out to be faulty. It will take a while for researchers to comb through Zhang’s argument to see whether it is correct.

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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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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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January 19, 2022

Quantum zeta epiphany: Physicist finds a new approach to a $1 million mathematical enigma

by University of California - Santa Barbara

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 analog 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 analog

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 analogs 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."

Journal information: Physical Review Letters

Provided by University of California - Santa Barbara

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Top Mathematician Claims He's Solved a $1 Million Hypothesis. But There's a Problem

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 September 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.

William Ross , Professor of Mathematics, University of Richmond .

This article is republished from The Conversation under a Creative Commons license. Read the original article .

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 Conjecture Solved 2022, published in a peer-reviewed journal, Link: http://www.sapub.org/global/showpaperpdf.aspx?doi=10.5923/j.ijtmp.20221202.03

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Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function.

The following are excluded:

Books by mathematical cranks (especially books by amateurs who claim to prove or disprove RH in their book)

Books about prime numbers or analytic number theory in general that include some material about the Riemann Hypothesis or Riemann's Zeta Function

Books that consist of collections of mathematical tables

Books that are paper-length (say, under 50 pages)

Doctoral dissertations (published books based upon doctoral dissertations are, of course, included)

• number-theory
• reference-request
• analytic-number-theory
• riemann-zeta
• riemann-hypothesis
• 7 $\begingroup$ I wonder if it would fit protocols better to post this as an answer after posting a short question that it answers. $\endgroup$ –  Michael Hardy Commented Feb 6, 2013 at 17:33
• 4 $\begingroup$ That will be a long list... consider writing it up as a BIBTeX bibliography. $\endgroup$ –  vonbrand Commented Feb 6, 2013 at 17:34
• 1 $\begingroup$ @vonbrand There are probably a few books missing, but I doubt more than 5-10 at most. I have been collecting books about this topic for years and own copies of all the books on my list except for the two by Laurincikas as I cannot find reasonably priced copies of them. One book I could have included but chose not to is Infirmation de l'hypothèse de Riemann by Henri Berliocchi, who is a respected French economist but apparently claims to disprove RH in the book. $\endgroup$ –  Marko Amnell Commented Feb 6, 2013 at 18:46
• $\begingroup$ @MarkoAmnell: I am making this Community Wiki . If you have some reason that this question should not be CW, flag this question for moderator attention. $\endgroup$ –  robjohn ♦ Commented Feb 6, 2013 at 19:41
• 1 $\begingroup$ I added István Sándor Gál's Lectures on algebraic and analytic number theory; with special emphasis on the theory of the Zeta functions of number fields and function fields to the list. The contents are described as follows: "Lectures given at Yale University and repeated at the University of Minnesota ... 1959-60 and 1960-61, respectively." $\endgroup$ –  Marko Amnell Commented Apr 18, 2015 at 11:09

2 Answers 2

Some of these are paper-length, not book-length, but they come up when I search Math Reviews for books, and who am I to argue with Math Reviews?

MR2934277 Reviewed van der Veen, Roland; van de Craats, Jan De Riemann-hypothese. (Dutch) [The Riemann hypothesis] Een miljoenenprobleem. [A million dollar problem] Epsilon Uitgaven, Utrecht, 2011. vi+102 pp. ISBN: 978-90-5041-126-4

MR2198605 Reviewed Jandu, Daljit S. The Riemann hypothesis and prime number theorem. Comprehensive reference, guide and solution manual. Infinite Bandwidth Publishing, North Hollywood, CA, 2006. 188 pp. ISBN: 0-9771399-0-5 11M26 (11N05) [From the publisher's description: "The author adopts the real analysis and technical basis to guide and solve the problem based on high school mathematics.''] [This one may not pass the "crank" test...]

MR1332493 Reviewed Ramachandra, K. On the mean-value and omega-theorems for the Riemann zeta-function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 85. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1995. xiv+169 pp. ISBN: 3-540-58437-4

MR1230387 Reviewed Ivić, A. Lectures on mean values of the Riemann zeta function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 82. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1991. viii+363 pp. ISBN: 3-540-54748-7

MR0747304 Reviewed van de Lune, J. Some observations concerning the zero-curves of the real and imaginary parts of Riemann's zeta function. Afdeling Zuivere Wiskunde [Department of Pure Mathematics], 201. Mathematisch Centrum, Amsterdam, 1983. i+25 pp.

MR0683287 Reviewed Klemmt, Heinz-Jürgen Asymptotische Entwicklungen für kanonische Weierstraßprodukte und Riemanns Überlegungen zur Nullstellenanzahl der Zetafunktion. (German) [Asymptotic expansions for canonical Weierstrass products and Riemann's reflections on the number of zeros of the zeta function] Nachrichten der Akademie der Wissenschaften in Göttingen II: Mathematisch-Physikalische Klasse 1982 [Reports of the Göttingen Academy of Sciences II: Mathematics-Physics Section 1982], 4. Akademie der Wissenschaften in Göttingen, Göttingen, 1982. 24 pp.

MR0637204 Reviewed van de Lune, J.; te Riele, H. J. J.; Winter, D. T. Rigorous high speed separation of zeros of Riemann's zeta function. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics], 113. Mathematisch Centrum, Amsterdam, 1981. ii+35 pp. (loose errata).

MR0541033 Reviewed te Riele, H. J. J. Tables of the first 15000 zeros of the Riemann zeta function to 28 significant digits, and related quantities. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics], 67. Mathematisch Centrum, Amsterdam, 1979. 155 pp. (not consecutively paged).

MR0565985 Reviewed van de Lune, J. On a formula of van der pol and a problem concerning the ordinates of the non-trivial zeros of Riemann's zeta function. Mathematisch Centrum, Afdeling Zuivere Wiskunde, ZW 16/73. Mathematisch Centrum, Amsterdam, 1973. iii+21 pp.

MR0359258 Reviewed \cyr Voĭtovich, N. N.; \cyr Nefedov, E. I.; \cyr Fialkovskiĭ, A. T. \cyr Pyatiznachnye tablitsy obobshchennoĭ dzeta-funktsii Rimana ot kompleksnogo argumenta. (Russian) [Five-place tables of the generalized Riemann zeta-function of a complex argument] With an English preface. Izdat. ``Nauka'', Moscow, 1970. 191 pp.

MR0266875 Reviewed Gavrilov, N. I. \cyr Problema Rimana o raspredelenii korneĭdzetafunktsii. (Russian) [The Riemann problem on the distribution of the roots of the zeta function ] Izdat. Lʹvov. Univ., Lvov, 1970 1970 172 pp.

MR0117905 Reviewed Haselgrove, C. B.; Miller, J. C. P. Tables of the Riemann zeta function. Royal Society Mathematical Tables, Vol. 6 Cambridge University Press, New York 1960 xxiii+80 pp.

• $\begingroup$ Thanks. The two books that stand out are the ones by Ramachandra and Ivic. The rest seem to be paper-length, collections of tables or in languages I cannot read. Ivic's book seems to be out of print. While looking for copies on Amazon, I stumbled on another book: Ramanujan Lecture Notes Series, Vol. 2: The Riemann zeta function and related themes: Proceedings of the international conference held at the National Institute of Advanced Studies, Bangalore, December 2003 . If one includes conference proceedings, there are probably more like that one. $\endgroup$ –  Marko Amnell Commented Feb 7, 2013 at 5:37
• 1 $\begingroup$ @MarkoAmnell: In the meantime, Ivić's book has become available on Kindle: amazon.com/… . $\endgroup$ –  joriki Commented Apr 16, 2020 at 16:04

These are all the books I am aware of that meet the criteria I set:

This list is available as a BibTeX bibliography file which can be downloaded from: http://drive.google.com/file/d/1HENOMh-Va368-vpKsI58mHVaGlm13ZaF/view?usp=sharing

Bernoulli Numbers and Zeta Functions , by Tsuneo Arakawa, Tomoyoshi Ibukiyama, Masanobu Kaneko, and Don B. Zagier, Springer (June 30, 2014), 274 pp.

The Riemann Hypothesis and the Distribution of Prime Numbers , by Naji Arwashan, Nova Science Pub Inc (April 15, 2021), 219 pp.

The Riemann Hypothesis - A Twenty-three centuries-long journey in search of the secret of prime numbers, Vol. 1 , by Jose Luis Perez Baeza, Parerga Foundation (Calle Major de Sarrià 232 PB, Barcelona 08017 ES), January 1, 2020, ISBN 978-8409257478, 493 pp.

Ramanujan Lecture Notes Series, Vol. 2: The Riemann zeta function and related themes (Proceedings of the international conference held at the National Institute of Advanced Studies, Bangalore, December 2003), R. Balasubramanian, K. Srinivas (Eds.), 206 pp.

The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike , Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller (Eds.), Springer, 2008

Equivalents of the Riemann Hypothesis , by Kevin Broughan, 3 volumes [Vol. 1: Arithmetic Equivalents , 400 pages; Vol. 2: Analytic Equivalents , 350 pages], Cambridge University Press (January 31, 2018); Vol. 3: Further Steps towards Resolving the Riemann Hypothesis , 704 pages (September 30, 2023)

Lectures on the Riemann zeta-function , by K. Chandrasekharan, Tata Institute of Fundamental Research, 1953, 148 pp.

The Riemann Hypothesis and Hilbert's Tenth Problem , by S. Chowla, Gordon and Breach, Science Publishers, Ltd., 1965

The Bloch-Kato Conjecture for the Riemann Zeta Function , John Coates, A. Raghuram, Anupam Saikia, R. Sujatha (Eds.), London Mathematical Society Lecture Note Series (Book 418), Cambridge University Press (April 30, 2015), 320 pp.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics , by John Derbyshire, Joseph Henry Press, 2003

Reassessing Riemann's Paper: On the Number of Primes Less Than a Given Magnitude , by Walter Dittrich, Springer (August 1, 2018), 65 pp.

The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics , by Marcus du Sautoy, HarperCollins, 2003

Riemann's Zeta Function , by Harold M. Edwards, Academic Press, 1974

Elizalde, Emilio, Ten Physical Applications of Spectral Zeta Functions , Lecture Notes in Physics 855, Springer, Berlin, 2012 (2nd edition), 290 pages

Elizalde, Emilio, Sergei D. Odintsov, August Romeo, A.A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications , World Scientific Publishing Company (1994), 336 pp.

Gál, István Sándor, Lectures on algebraic and analytic number theory; with special emphasis on the theory of the Zeta functions of number fields and function fields , Jones Letter Service, Minneapolis, 1961, 453 pp.

Gavrilov, N. I. Problema Rimana o raspredelenii korneidzetafunktsii . (Russian) [The Riemann problem on the distribution of the roots of the zeta function] Izdat. L'vov. Univ., Lvov, 1970 172 pp.

Simply Riemann (Great Lives) , by Jeremy Gray, Simply Charly (March 20, 2020), 167 pp.

The Mysteries of the Real Prime , by M.J. Shai Haran, London Mathematical Society (December 6, 2001), 256 pp.

The Riemann hypothesis in algebraic function fields over a finite constants field , by Helmut Hasse, Dept. of Mathematics, Pennsylvania State University, 1968, 235 pp. [Verbatim reproduction of lectures given at Pennsylvania State University, Spring term, 1968]

Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality , by Hafedh Herichi, World Scientific Pub Co Inc (July 31, 2019), 400 pp.

Ivic, A. Lectures on mean values of the Riemann zeta function . Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 82. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1991. viii+363 pp. ISBN: 3-540-54748-7

The Riemann Zeta-Function: Theory and Applications , by Aleksandar Ivic, John Wiley & Sons, Inc., 1985

Ivic, A. The Theory of Hardy's Z-function . Cambridge Tracts in Mathematics 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8, 264 pages, 2012

Ivic, A. Topics in recent zeta function theory . Publ. Math. d'Orsay, Université de Paris-Sud, Dép. de Mathématique, 1983, 272 pages

Lectures on the Riemann Zeta Function , by H. Iwaniec, American Mathematical Society (October 30, 2014), 119 pp.

Contributions to the Theory of Zeta-Functions: The Modular Relation Supremacy , by Shigeru Kanemitsu and Haruo Tsukada, World Scientific Publishing Company (June 30, 2014), 280 pp.

The Riemann Zeta-Function , by Anatoly A. Karatsuba and S. M. Voronin, Walter de Gruyter & Co., 1992

Random Matrices, Frobenius Eigenvalues, and Monodromy , by Nicholas M. Katz and Peter Sarnak, American Mathematical Society (November 24, 1998), 419 pp.

Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions , by Michel L. Lapidus and Machiel van Frankenhuysen, Birkhäuser, 1999

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings , by Michel L. Lapidus and Machiel van Frankenhuysen, Springer, 2006

In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes , by Michel L. Lapidus, American Mathematical Society, 2008

Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions , by Michel L. Lapidus, Goran Radunović and Darko Žubrinić, Springer (February 1, 2017), 704 pp.

Limit Theorems for the Riemann Zeta-Function , by Antanas Laurincikas, Kluwer Academic Publishers, 1996

The Lerch zeta-function , by Antanas Laurincikas and Ramunas Garunkstis, Kluwer Academic Publishers, 2002

Recent Progress on Topics of Ramanujan Sums and Cotangent Sums Associated with the Riemann Hypothesis , by Helmut Maier, Laszlo Toth and Michael Th. Rassias, World Scientific Publishing Co Pte Ltd (March 10, 2022), 180 pp.

Prime Numbers and the Riemann Hypothesis , by Barry Mazur and William Stein, Cambridge University Press (October 31, 2015), 150 pp.

Exploring the Riemann Zeta Function: 190 years from Riemann's Birth , Hugh Montgomery, Ashkan Nikeghbali, Michael Th. Rassias (Eds.), Springer (September 9, 2017), 272 pp.

Spectral Theory of the Riemann Zeta-Function , by Yoichi Motohashi, Cambridge University Press, 1997

A Study of Bernhard Riemann's 1859 Paper , by Terrence P. Murphy, Paramount Ridge Press (September 18, 2020), 182 pp.

In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem , by Paul J. Nahin, Princeton University Press (October 19, 2021), 344 pp.

An Introduction to the Theory of the Riemann Zeta-Function , by S. J. Patterson, Cambridge University Press, 1988

Ramachandra, K. On the mean-value and omega-theorems for the Riemann zeta-function . Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 85. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1995. xiv+169 pp. ISBN: 3-540-58437-4

The Theory of the Hurwitz Zeta Function of the Second Variable , by Vivek V. Rane, Alpha Science International Ltd (December 31, 2015), 300 pp.

Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers , by Dan Rockmore, Random House, Inc., 2005

The Riemann Hypothesis in Characteristic p in Historical Perspective , by Peter Roquette, Springer (September 30, 2018), 300 pp.

The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics , by Karl Sabbagh, Farrar, Straus, and Giroux, 2002

History of Zeta Functions , by Robert Spira, 3 volumes, Quartz Press (392 Taylor Street, Ashland OR 97520-3058), 1218 pages, 1999, ISBN 0-911455-10-8

Seminar on the Riemann Zeta Function 1965-1966 , by Robert Spira, Mimeographed typescript, University of Tennessee, Knoxville, 57 pages

Zeta and q-Zeta Functions and Associated Series and Integrals , by H. M. Srivastava and Junesang Choi, Elsevier Inc., 2012

New Directions in Value-distribution Theory of Zeta and L-functions: Wurzburg Conference, October 6-10, 2008 (Berichte aus der Mathematik), Rasa Steuding, Jörn Steuding (Eds.), Shaker Verlag GmbH, Germany (December 31, 2009), 346 pp.

Bohr-Jessen Limit Theorem, Revisited , by Satoshi Takanobu, Mathematical Society of Japan Memoirs (Book 31), Mathematical Society of Japan (July, 2013), 216 pp.

Zeta and eta functions: A new hypothesis , by Ashwani Kumar Thukral, CreateSpace Independent Publishing Platform (December 17, 2015), 56 pp.

The Theory of the Riemann Zeta-Function , by E. C. Titchmarsh, D. R. Heath-Brown (Ed.), Second edition, Oxford University Press, 1986

Pseudodifferential Methods in Number Theory , by André Unterberger, Birkhäuser (July 24, 2018), 180 pages

Van der Veen, Roland; van de Craats, Jan De Riemann-hypothese . (Dutch) [The Riemann hypothesis] Een miljoenenprobleem . [A million dollar problem] Epsilon Uitgaven, Utrecht, 2011. vi+102 pp. ISBN: 978-90-5041-126-4

The Riemann Hypothesis , by Roland van der Veen and Jan van de Craats, The Mathematical Association of America (January 6, 2016), 154 pp.

Van Frankenhuijsen, Machiel, The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators , London Mathematical Society Student Texts (Book 80), Cambridge University Press (January 9, 2014), 162 pp.

Zeta Functions over Zeros of Zeta Functions , by André Voros, Springer-Verlag, 2010

Zeta Functions of Reductive Groups and Their Zeros , by Lin Weng, World Scientific Publishing Co Pte Ltd (May 19, 2018), 550 pp.

• $\begingroup$ you should classify them, putting the Titchmarsh in good place (but say that there is nothing in it about Dirichlet characters, number fields, nor automorphic forms) and maybe adding a book on number fields, and another on automorphic forms ? $\endgroup$ –  reuns Commented May 27, 2016 at 0:54
• $\begingroup$ @user1952009: That would be a different (and perhaps worthwhile) project. If I started to add books about various topics in number theory (or related fields) which aren't devoted entirely (or mainly) to the Riemann Hypothesis or the Riemann zeta function, the list would quickly grow to be much longer, and it would be very hard to decide what books to include, and what to exclude. Similarly, classifying the books in some way, or adding descriptions of their contents, is yet another different project. My question (and the resulting list) is defined more narrowly. Even now, decisions about... $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 3:37
• $\begingroup$ (continued): whether to include a particular book can be open to debate. For example, I excluded Marcus du Sautoy's Zeta Functions of Groups and Rings because it doesn't seem to me to be directly relevant to RH but I freely admit I could turn out to be wrong about that. But I included Random Matrices, Frobenius Eigenvalues, and Monodromy by Nicholas Katz and Peter Sarnak because Sarnak himself says he thinks the ideas in that book will be crucial to finding a proof of RH. See e.g. math.stackexchange.com/questions/327693/… $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 3:47
• $\begingroup$ in my opinion it is not "various topics in number theory" but only some of the main aspects of the problem www.claymath.org/sites/default/files/official_problem_description.pdf . and do you have pdf copies of all those books ? $\endgroup$ –  reuns Commented May 29, 2016 at 14:06
• $\begingroup$ Fair enough, but how would you decide which books about number fields, or automorphic forms, to include? All of them? I actually own printed copies of all the books on my list except for five, and I will hopefully acquire one more in a week or two if Vivek Rane's The Theory of the Hurwitz Zeta Function of the Second Variable is finally published on May 31 after several delays. $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 17:14

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Monographs in Number Theory: Volume 10

Recent progress on topics of ramanujan sums and cotangent sums associated with the riemann hypothesis.

• By (author): 
• Helmut Maier ( University of Ulm, Germany ) , 
• Michael Th Rassias ( Hellenic Military Academy, Greece ) , and 
• László Tóth ( University of Pécs, Hungary )
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• Description
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In this monograph, we study recent results on some categories of trigonometric/exponential sums along with various of their applications in Mathematical Analysis and Analytic Number Theory. Through the two chapters of this monograph, we wish to highlight the applicability and breadth of techniques of trigonometric/exponential sums in various problems focusing on the interplay of Mathematical Analysis and Analytic Number Theory. We wish to stress the point that the goal is not only to prove the desired results, but also to present a plethora of intermediate Propositions and Corollaries investigating the behaviour of such sums, which can also be applied in completely different problems and settings than the ones treated within this monograph.

In the present work we mainly focus on the applications of trigonometric/exponential sums in the study of Ramanujan sums — which constitute a very classical domain of research in Number Theory — as well as the study of certain cotangent sums with a wide range of applications, especially in the study of Dedekind sums and a facet of the research conducted on the Riemann Hypothesis. For example, in our study of the cotangent sums treated within the second chapter, the methods and techniques employed reveal unexpected connections with independent and very interesting problems investigated in the past by R de la Bretèche and G Tenenbaum on trigonometric series, as well as by S Marmi, P Moussa and J-C Yoccoz on Dynamical Systems.

Overall, a reader who has mastered fundamentals of Mathematical Analysis, as well as having a working knowledge of Classical and Analytic Number Theory, will be able to gradually follow all the parts of the monograph. Therefore, the present monograph will be of interest to advanced undergraduate and graduate students as well as researchers who wish to be informed on the latest developments on the topics treated.

Sample Chapter(s) Preface Chapter 1: Ramanujan Sums: A Survey of Recent Results

• Preliminaries
• Arithmetic Functions of One and Several Variables
• Basic Properties of Ramanujan Sums
• Sums of Products of Ramanujan Sums
• Weighted Averages of Ramanujan Sums
• Properties of Even Functions (mod r )
• Counting Solutions of Congruences in Several Variables
• Polynomials of Which Coefficients are Ramanujan Sums
• Analogs and Generalizations of Ramanujan Sums
• Ramanujan Expansions of Arithmetic Functions
• Cotangent Sums Related to the Estermann Zeta Function and to the Riemann Hypothesis
• The Distribution of Cotangent Sums Related to the Estermann Zeta Function
• Moments of Cotangent Sums Related to the Estermann Zeta Function
• The Order of Magnitude for Moments for Certain Cotangent Sums
• The Distribution of Cotangent Sums for Arguments from Special Sequences and Joint Distribution for Various Arguments
• Results Related to the Nyman–Beurling Criterion for the Riemann Hypothesis
• The Maximum of Cotangent Sums for Rational Numbers in Short Intervals
• Open Problems
• Bibliography

Readership: Scientists, researchers, and graduate students in Pure and Applied Mathematics.

FRONT MATTER

• Pages: i–ix

https://doi.org/10.1142/9789811246890_fmatter

Chapter 1: Ramanujan Sums: A Survey of Recent Results

• Pages: 1–64

https://doi.org/10.1142/9789811246890_0001

• Periodic functions (mod r )
• Even functions (mod r )
• The DFT of even functions (mod r )
• Menon-type identities concerning Dirichlet characters
• Menon-type identities concerning additive characters
• Linear congruences with constraints
• Quadratic congruences
• Unitary Ramanujan sums
• Modified unitary Ramanujan sums
• Ramanujan sums defined by regular systems of divisors
• Ramanujan sums defined by regular integers (mod n )
• Expansions of functions with respect to classical and unitary Ramanujan sums
• Expansions of functions with respect to modified unitary Ramanujan sums

Chapter 2: Cotangent Sums Related to the Estermann Zeta Function and to the Riemann Hypothesis

• Pages: 65–144

https://doi.org/10.1142/9789811246890_0002

In this chapter, we give an overview of results of H. Maier and M. Th. Rassias on various aspects of the following cotangent sum: c 0 ( r b ) : = − ∑ m = 1 b − 1 m b cot ( π m r b ) , ( * ) where r, b ∈ ℕ, b ≥ 2, 1 ≤ r ≤ b and ( r , b ) =1…

BACK MATTER

• Pages: 145–154

https://doi.org/10.1142/9789811246890_bmatter

Helmut Maier is a Professor at the University of Ulm, Germany, since 1993. Maier received his PhD in 1981 by the University of Minnesota, USA, under the supervision of Professor J Ian Richards. The thesis was an extension of his paper Chains of large gaps between consecutive primes , in which Maier for the first time applied what is now called Maier's Matrix Method . This method later on led him and other mathematicians to the discovery of unexpected irregularities in the distribution of prime numbers.

Before becoming a Professor at the Mathematics Department of the University of Ulm, Maier held the position of an Assistant Professor and later an Associate Professor at the University of Georgia, USA. He has published a plethora of deep papers in Analytic Number Theory.

Michael Th. Rassias is an Associate Professor at the Department of Mathematics and Engineering Sciences of the Hellenic Military Academy, a visiting Researcher at the Institute for Advanced Study, Princeton, as well as a visiting Associate Professor at the Moscow Institute of Physics and Technology. He obtained his PhD in Mathematics from ETH-Zürich in 2014. During the academic year 2014–2015, he was a Postdoctoral researcher at the Department of Mathematics of Princeton University and the Department of Mathematics of ETH-Zürich, conducting research at Princeton. While at Princeton, he prepared with John F Nash, Jr. the volume Open Problems in Mathematics , Springer, 2016. He has received several awards in mathematical problem-solving competitions, including a Silver medal at the International Mathematical Olympiad of 2003 in Tokyo. He has authored and edited several books. His current research interests lie in mathematical analysis, analytic number theory, and more specifically the Riemann Hypothesis, Goldbach's conjecture, the distribution of prime numbers, approximation theory, functional equations and analytic inequalities.

László Tóth is a Professor and Head of Department at the University of Pécs, Hungary. He obtained the PhD degree in Mathematics at the Babeș-Bolyai University of Cluj, Romania in 1996, as well as at the University of Debrecen, Hungary in 2003, and Habilitation in Mathematics at the University of Debrecen in 2006. He received the title Doctor of the Hungarian Academy of Sciences in 2020. He has published more than one hundred scientific papers in Number Theory, Combinatorics and Group Theory. He is editor and referee for several mathematical journals, and Editor-in-Chief of the journal Mathematica Pannonica .

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Cotangent sums play a significant role in the Nyman–Beurling criterion for the Riemann hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

Helmut Maier. Michael Th. Rassias. "On the maximum of cotangent sums related to the Riemann hypothesis in rational numbers in short intervals." Mosc. J. Comb. Number Theory 10 (4) 303 - 313, 2022. https://doi.org/10.2140/moscow.2021.10.303

Information

MathSciNet: MR4366117 Digital Object Identifier: 10.2140/moscow.2021.10.303

Subjects: Primary: 11L03 , 11M06 , 26A12

Keywords: cotangent sums , Estermann’s zeta function , Kloosterman sums , Nyman–Beurling criterion , Riemann hypothesis , Riemann zeta function

Rights: Copyright © 2022 Mathematical Sciences Publishers

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