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Studying the ‘Music of the Primes’

  • http-blogs-fandm-edu-wp-content-blogs-dir-29-files-2012-04-tyoder-jpg
  • Ted Yoder ’12

Ted Yoder ’12 is the 2011 winner of the Student Mathematical Paper Competition sponsored by the Eastern Pennsylvania and Delaware chapter of the Mathematical Association of America.

Ted Yoder ’12, a mathematics and physics double major at Franklin & Marshall College, has won the 2011 Student Mathematical Paper Competition sponsored by the Eastern Pennsylvania and Delaware chapter of the Mathematical Association of America. Yoder’s winning entry was titled “An Introduction to the Riemann Hypothesis.”

In writing about the Riemann Hypothesis, Yoder addressed the most famous unsolved problem in mathematics to make it accessible to an undergraduate audience. He wrote the paper during his Number Theory class taught by Associate Professor of Mathematics Wendell Ressler last year.

“Bernhard Riemann proposed this hypothesis in 1859, but there wasn’t much indication of why he thought this,” says Yoder, who received a Goldwater Scholarship in 2010. “If the hypothesis is true, the prime numbers are distributed in the most beautiful way possible. You hear mathematicians talk about the ‘music of the primes.’ That’s at the heart of this problem. People are interested in this problem because of its relation to the primes.”

The Riemann Hypothesis addresses the distribution of prime numbers—those that cannot be expressed as the product of two smaller ones. The distribution of primes among all numbers does not follow a regular pattern, although Riemann attempted to explain the distribution with the Riemann zeta function. Yoder reported on the problem by defining the Riemann zeta function, then providing an explanation of Riemann's analytic continuation and its use to show that the zeta function zeros are restricted to the critical strip in the complex plane.

The Reimann Hypothesis is one of seven Millenium Prize Problems established by the Clay Mathematics Institute in 2000; its proof would earn a reward of $1,000,000.

“The problem Ted wrote about involves complex analysis,” Ressler says. “It’s a tough problem for undergrads to think about and write about. Ted’s a really good student and did a marvelous job making it clear and understandable. It was a no-brainer to suggest that he submit the paper to the competition.”

Eric Kahn, assistant professor of mathematics at Bloomsburg University, informed Yoder of his winning entry earlier this semester. In the award letter, Kahn said the paper was “a delightful presentation of a famously unsolved problem which you made accessible at the undergraduate level.”

After spending his first two summers at F&M as a Hackman Scholar, Yoder participated in a Research Experiences for Undergraduates (REU) program at Bucknell University this past summer. He is currently applying to graduate schools, and hopes to become a professor or research scientist.

With one impressive paper under his belt, others will surely follow for the Honors List student. Yoder hopes his work will demonstrate the beauty of mathematics. 

“It’s important to take away from the Riemann Hypothesis that math can have deep consequences that lead to elegant and beautiful results,” Yoder says.