Mathematicians revive abandoned approach to the Riemann Hypothesis

The idea for the paper was sparked by a "toy problem" that Emory mathematician Ken Ono (left) presented as a "gift" to entertain Don Zagier (right), of the Max Planck Institute of Mathematics, to celebrate Zagier's 65th birthday. The toy problem is seen on the whiteboard behind them.

By Carol Clark

Many ways to approach the Riemann Hypothesis have been proposed during the past 150 years, but none of them have led to conquering the most famous open problem in mathematics. A new paper in the Proceedings of the National Academy of Sciences (PNAS) suggests that one of these old approaches is more practical than previously realized.

“In a surprisingly short proof, we’ve shown that an old, abandoned approach to the Riemann Hypothesis should not have been forgotten,” says Ken Ono, a number theorist at Emory University and co-author of the paper. “By simply formulating a proper framework for an old approach we’ve proven some new theorems, including a large chunk of a criterion which implies the Riemann Hypothesis. And our general framework also opens approaches to other basic unanswered questions.”

The paper builds on the work of Johan Jensen and George Pólya, two of the most important mathematicians of the 20th century. It reveals a method to calculate the Jensen-Pólya polynomials — a formulation of the Riemann Hypothesis — not one at a time, but all at once.

“The beauty of our proof is its simplicity,” Ono says. “We don’t invent any new techniques or use any new objects in math, but we provide a new view of the Riemann Hypothesis. Any reasonably advanced mathematician can check our proof. It doesn’t take an expert in number theory.”

Read a commentary on the paper by Fields Medalist Enrico Bombieri.

Although the paper falls short of proving the Riemann Hypothesis, its consequences include previously open assertions which are known to follow from the Riemann Hypothesis, as well as some proofs of conjectures in other fields.

Co-authors of the paper are Michael Griffin and Larry Rolen — two of Ono’s former Emory graduate students who are now on the faculty at Brigham Young University and Vanderbilt University, respectively — and Don Zagier of the Max Planck Institute of Mathematics.

“The result established here may be viewed as offering further evidence toward the Riemann Hypothesis, and in any case, it is a beautiful stand-alone theorem,” says Kannan Soundararajan, a mathematician at Stanford University and an expert on the Riemann Hypothesis.

"Math at a research level is often more art than calculation and that was certainly the case here," says co-author Michael Griffin, an Emory grad who is now on the faculty at Brigham Young University.

The idea for the paper was sparked two years ago by a “toy problem” that Ono presented as a “gift” to entertain Zagier during the lead-up to a math conference celebrating his 65th birthday. A toy problem is a scaled-down version of a bigger, more complicated problem that mathematicians are trying to solve. Zagier described the one that Ono gave him as “a cute problem about the asymptotic behavior of certain polynomials involving Euler’s partition function, which is an old love of mine and of Ken’s — and of about pretty much any classical number theorist.”

“I found the problem intractable and I didn’t really expect Don to get anywhere with it,” Ono recalls. “But he thought the challenge was super fun and soon he had crafted a solution.”

Ono’s hunch was that such a solution could be crafted into a more general theory. That’s what the mathematicians ultimately achieved.

“It’s been a fun project to work on, a really creative process,” Griffin says. “Math at a research level is often more art than calculation and that was certainly the case here. It required us to look at an almost 100-year-old idea of Jensen and Pólya in a new way.”

The Riemann Hypothesis is one of seven Millennium Prize Problems, identified by the Clay Mathematics Institute as the most important open problems in mathematics. Each problem carries a $1 million bounty for its solvers.

The method outlined in the PNAS paper "has a shocking sense of being universal, in that it applies to problems that are seemingly unrelated," says co-author Larry Rolen, an Emory grad now on the faculty at Vanderbilt University.

The hypothesis debuted in an 1859 paper by German mathematician Bernhard Riemann. He noticed that the distribution of prime numbers is closely related to the zeros of an analytical function, which came to be called the Riemann zeta function. In mathematical terms, the Riemann Hypothesis is the assertion that all of the nontrivial zeros of the Zeta function have real part ½.

“His hypothesis is a mouthful, but Riemann’s motivation was simple,” Ono says. “He wanted to count prime numbers.”

Bernhard Riemann

The hypothesis is a vehicle to understand one of the greatest mysteries in number theory — the pattern underlying prime numbers. Although prime numbers are simple objects defined in elementary math (any number greater than 1 with no positive divisors other than 1 and itself) their distribution remains hidden.

The first prime number, 2, is the only even one. The next prime number is 3, but primes do not follow a pattern of every third number. The next is 5, then 7, then 11. As you keep counting upwards, prime numbers rapidly become less frequent.

“It’s well known that there are infinitely many prime numbers, but they become rare, even by the time you get to the 100s,” Ono explains. “In fact, out of the first 100,000 numbers, only 9,592 are prime numbers, or roughly 9.5 percent. And they rapidly become rarer from there. The probability of picking a number at random and having it be prime is zero. It almost never happens.”

In 1927, Jensen and Pólya formulated a criterion for confirming the Riemann Hypothesis, as a step toward unleashing its potential to elucidate the primes and other mathematical mysteries. The problem with the criterion — establishing the hyperbolicity of the Jensen-Pólya polynomials — is that it is infinite. During the past 90 years, only a handful of the polynomials in the sequence have been verified, causing mathematicians to abandon this approach as too slow and unwieldy.

For the PNAS paper, the authors devised a conceptual framework that combines the polynomials by degrees. This method enabled them to confirm the criterion for each degree 100 percent of the time, eclipsing the handful of cases that were previously known.

“The method has a shocking sense of being universal, in that it applies to problems that are seemingly unrelated,” Rolen says. “And at the same time, its proofs are easy to follow and understand. Some of the most beautiful insights in math are ones that took a long time to realize, but once you see them, they appear simple and clear.”

Despite their work, the results don’t rule out the possibility that the Riemann Hypothesis is false and the authors believe that a complete proof of the famous conjecture is still far off.

The work was supported by grants from the National Science Foundation and the Asa Griggs Candler Fund.

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Artificial intelligence and 'deep ethics'

In the sci-fi film “2001: A Space Odyssey,” astronauts go into a soundproof pod to discuss their concerns about some of the decisions made by the supercomputer Hal (seen through the window) without realizing that Hal knows how to lip read.

Advances in neurotechnology, genetics and artificial intelligence are not only going to change society as a whole, they are actually going to challenge what it means to be human and change our ethics, argues Paul Root Wolpe, director of the Emory Center for Ethics, in a recent TEDx Atlanta talk.

He uses self-driving cars as just one example.

“These vehicles are going to be going down the road and in a crisis they’re going to have to make decisions about what to do,” Wolpe says. “Do I crash into the wall and endanger my passengers or do I turn left and hit those pedestrians? For the first time we’re going to have to create ethical algorithms. That is, we’re going to have to teach a vehicle to make ethical decisions. For the first time, machines will be making ethical decisions that will have a profound impact on human beings.” 

Watch Wolpe’s talk in the video below to learn what he means by the term “deep ethics,” and how artificial intelligence may someday help us navigate through the ethical complexities raised by technology itself.



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Ginkgo seed extracts show antibacterial activity on skin pathogens

The fan-shaped leaves of a ginkgo tree on the Emory campus, just off the Quad on the west end of Carlos Hall. The tree led a student to an ancient Chinese text and a laboratory discovery. (Photo by Ann Watson)

Extracts from the seeds of the Ginkgo biloba tree show antibacterial activity on pathogens that can cause skin infections such as acne, psoriasis, dermatitis and eczema, a study at Emory University finds. Frontiers in Microbiology is publishing the results of laboratory experiments showing that the extracts inhibit the growth of Cutibacterium acnes, Staphylococcus aureus and Streptococcus pyogenes.

A nearly 200-year-old copy of a 16th-century text on traditional Chinese medicine, the Ben Cao Gang Mu, guided the researchers in their experiments. “It was like blowing the dust off knowledge from the past and rediscovering something that had been there all along,” says Xinyi (Xena) Huang, co-first author of the paper.

Huang, a native of China, began the project for her senior thesis as a biology major at Emory. She has since graduated from Emory and is now a student at the University of Maryland School of Pharmacy.

“To the best of our knowledge, this is the first study to demonstrate the antibacterial activity of ginkgo seeds on skin pathogens,” says Cassandra Quave, senior author of the paper and assistant professor at Emory’s Center for the Study of Human Health and the School of Medicine’s Department of Dermatology. “This paper is just one more example of how much we still have to learn about the pharmacological potential of the complex chemistry of plants.”

Quave is an ethnobotanist, studying how indigenous people use plants in their healing practices, to uncover promising candidates for new drugs. Read more about the discovery.

Click here to read more about the discovery.

When do children alter behavior to please others?

“I have spent the past four years at Emory University investigating how an infant, who has no problem walking around the grocery store in her onesie, develops into an adult that fears public speaking for fear of being negatively judged,” says Sara Botto in her newly released TEDxAtlanta talk.

Botto is a doctoral candidate in the Cognition and Development program of Emory’s Department of Psychology. Together with Emory psychologist Philippe Rochat, she designed experiments to investigate when in development we become sensitive to others’ evaluations — a big part of being human.

Watch the TEDxAtlanta video below to see young children reacting to the opinions of others during the experiments, which take the form of a game called “The Robot Task.”

Botto’s research showed that, even before they can form a simple sentence, children are sensitive to the evaluations of others, and alter their behavior accordingly.

“Whether we’re aware of it or not, we’re constantly communicating values to others,” Botto says. “We’re communicating a value when we mostly compliment girls for their pretty hair or their pretty dress but boys for their intelligence. Or when we choose to offer candy as opposed to nutritious food as a reward for good behavior.”

Visit Botto’s web site, AdultingWithKids.com, to learn more about credible, science-based child development research.




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Gender gap in spatial reasoning starts in elementary school, meta-analysis finds

"We're interested in the origins of gender differences in spatial skills because of their potential role in the gender gap we see in math and science fields," says Jillian Lauer, who is set to graduate from Emory in May with a PhD in psychology. (Getty Images)

By Carol Clark

It is well-established that, on average, men outperform women on a spatial reasoning task known as mental rotation — imagining multi-dimensional objects from different points of view. Men are not, however, born with this advantage, suggests a major meta-analysis by psychologists at Emory University. Instead, males gain a slight advantage in mental-rotation performance during the first years of formal schooling, and this advantage slowly grows with age, tripling in size by the end of adolescence.

The Psychological Bulletin, a journal of the American Psychological Association, is publishing the findings.

“Some researchers have argued that there is an intrinsic gender difference in spatial reasoning — that boys are naturally better at it than girls,” says lead author Jillian Lauer, who is set to graduate from Emory in May with a PhD in psychology. “While our results don’t exclude any possibility that biological influences contribute to the gender gap, they suggest that other factors may be more important in driving the gender difference in spatial skills during childhood.”

Co-authors of the paper include Stella Lourenco, associate professor of psychology at Emory, whose lab specializes in the development of spatial and numerical cognition. Co-author Eukyung Yhang worked on the paper as an Emory undergraduate, funded by the university’s Institute for Quantitative Theory and Methods. Yhang graduated in 2018 and is now at Yale University School of Medicine.

The meta-analysis included 128 studies of gender differences in spatial reasoning, combining statistics on more than 30,000 children and adolescents aged three to 18 years. The authors found no gender difference in mental-rotation skills among preschoolers, but a small male advantage emerged in children between the ages of six and eight.

While differences in verbal and mathematical abilities between men and women tend to be small or non-existent, twice as many men as women are top performers in mental rotation, making it one of the largest gender differences in cognition.

Mental rotation is considered one of the hallmarks of spatial reasoning. “If you’re packing your suitcase and trying to figure out how each item can fit within that space, or you’re building furniture based on a diagram, you’re likely engaged in mental rotation, imagining how different objects can rotate to fit together,” Lauer explains.

It takes most of childhood and adolescence for the gender gap in spatial skills to reach the size of the difference seen in adulthood.

Prior research has also shown that superior spatial skills predict success in male-dominated science, technology engineering and math (STEM) fields, and that the gender difference in spatial reasoning may contribute to the gender disparity in these STEM fields.

“We’re interested in the origins of gender differences in spatial skills because of their potential role in the gender gap we see in math and science fields,” Lauer says. “By determining when the gender difference can first be detected in childhood and how it changes with age, we may be able to develop ways to make educational systems more equitable.”

It takes most of childhood and adolescence for the gender gap in spatial skills to reach the size of the difference seen in adulthood, Lauer says. She adds that the meta-analysis did not address causes for why the gender gap for mental rotation emerges and grows.

Lauer notes that previous research has shown that parents use more spatial language when they talk to preschool sons than daughters. Studies have also found that girls report more anxiety about having to perform spatial tasks than do boys by first grade, and that children are aware of gender stereotypes about spatial intelligence during elementary school.

“Now that we’ve characterized how gender differences in spatial reasoning skills develop in children over time we can start to hone in on the reasons for those differences,” Lauer says.

Meanwhile, she adds, parents may want to be aware to encourage both their daughters and sons to play with blocks and other construction items that might help in the development of spatial reasoning skills, since evidence shows that these skills can be improved with training.

“Giving both girls and boys more opportunities to develop their spatial skills is something that parents and educators have the power to do,” Lauer says.

Lauer has accepted a post-doctoral fellowship position at New York University. Her PhD advisor is Patricia Bauer, a professor of psychology at Emory focused on cognition and child development.

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