# UW scientists unlock major number theory puzzle

February 27, 2007MADISON-Mathematicians have finally laid to rest the legendary mystery surrounding an elusive group of numerical expressions known as the "mock theta functions."

Number theorists have struggled to understand the functions ever since the great Indian mathematician Srinivasa Ramanujan first alluded to them in a letter written on his deathbed, in 1920. Now, using mathematical techniques that emerged well after Ramanujan's death, two number theorists at the University of Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them.

Their new theory is proving invaluable in the resolution of long-standing open questions in number theory. In addition, the UW-Madison advance will for the first time enable researchers to apply mock theta functions to problems in a variety of fields, including physics, chemistry and several branches of mathematics. The findings appear in a series of three papers, the third appearing this week in the Proceedings of the National Academy of Sciences.

"It's extremely gratifying to be able to say we solved the 'final problem' of Ramanujan," says co-author Ken Ono, UW-Madison Manasse Professor of Letters and Science, who is widely noted for contributions to number theory. "We simply got really lucky."

Ono worked in collaboration with German mathematician Kathrin Bringmann, a postdoctoral researcher at UW-Madison.

"This is something I really didn't expect anybody to do," says George Andrews, a leading number theorist at Pennsylvania State University who in 2000 called mock theta functions one of the most difficult math puzzles of the new millennium. "It is an outstanding piece of work, a breathtakingly wonderful achievement."

Working from Ramanujan's letter, number theorists believed that mock theta functions are related to a well-understood class of mathematical expressions-the 'theta' functions--that have been in use for centuries. Theta functions constitute a certain sequence of numbers that has proved useful in various problems of mathematical analysis.

Mock theta functions similarly constitute an infinite series of numbers. But what has been completely baffling is what it is about mock theta series that make them so rich and powerful. Over the decades-much to the amazement of mathematicians everywhere--mock theta functions have cropped up amidst calculations in a number of fields, including mathematics, physics, chemistry, and even cancer research.

What made mock theta functions all the more inscrutable was the fact that the first few pages of Ramanujan's letter were lost. Those pages may have contained more clues, but in their absence, the letter merely presented 17 examples of the functions. What's missing is any definition of what the functions are, any hints on how to derive them, and any indication of why they are even important. All those secrets died with Ramanujan just two months after he wrote the letter, when he succumbed to tuberculosis at the age of 32.

"Imagine stringing together a thousand random words and then saying you've come up with the most beautiful poetry," says Ono. "That's essentially what Ramanujan did to us." Bringmann and Ono made sense of it all by finding a way to represent the power of mock theta functions through another relatively new family of mathematical expressions known as the Harmonic Maass Forms.

A Dutch mathematician named Sander Zwegers had already made that important connection in 2002, but he had focused only on Ramanujan's examples.

It was during a flight to New Hampshire that Ono realized the full depth and meaning of Zwegers' work. Skimming a journal to pass the time, Ono happened upon an old article by George Andrews on mock theta functions. Suddenly, he noticed that some of the mathematics in the paper seemed to resonate with parts of the Harmonic Maass theory, which he and Bringmann just happened to be developing at the time, for other reasons.

The mathematicians found the connection held up beautifully. "We knew we were onto something right away," says Ono. "It was an uncanny set of coincidences that lead us to this solution. It was as if it all just fell into our lap and now we are serendipitously applying our theory to longstanding open problems."

-end-

University of Wisconsin-Madison

**Related Mathematics Articles:**

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Mathematics supports a new way to classify viruses based on structure

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

The mathematics of coffee extraction: Searching for the ideal brew

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Even physicists are 'afraid' of mathematics

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Mathematics and music: New perspectives on the connections between these ancient arts

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

Kindergarteners' mathematics success hinges on preschool skills

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

First international mathematics research institute launched in Australia

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

**Related Mathematics Reading:**

## Best Science Podcasts 2019

We have hand picked the**best science podcasts**for 2019. Sit back and enjoy new science podcasts updated daily from your favorite science news services and scientists.

**Now Playing: TED Radio Hour**

**Don't Fear Math**

Why do many of us hate, even fear math? Why are we convinced we're bad at it? This hour, TED speakers explore the myths we tell ourselves and how changing our approach can unlock the beauty of math. Guests include budgeting specialist Phylecia Jones, mathematician and educator Dan Finkel, math teacher Eddie Woo, educator Masha Gershman, and radio personality and eternal math nerd Adam Spencer.

**Now Playing: Science for the People**

**#517 Life in Plastic, Not Fantastic**

Our modern lives run on plastic. It's in the computers and phones we use. It's in our clothing, it wraps our food. It surrounds us every day, and when we throw it out, it's devastating for the environment. This week we air a live show we recorded at the 2019 Advancement of Science meeting in Washington, D.C., where Bethany Brookshire sat down with three plastics researchers - Christina Simkanin, Chelsea Rochman, and Jennifer Provencher - and a live audience to discuss plastics in our oceans. Where they are, where they are going, and what they carry with them. Related links:...