Fractal Structure to Partition Function

January 20, 2011.   Researchers from Emory, the University of Wisconsin at Madison, Yale, and Germany's Technical University of Darmstadt discovered that partition numbers behave like fractals, possessing an infinitely-repeating structure.

In a collaborative effort sponsored by the American Institute of Mathematics and the National Science Foundation, a team of mathematicians led by Ken Ono developed new techniques to explore the nature of the partition numbers. "We prove that partition numbers are 'fractal' for every prime. Our 'zooming' procedure resolves several open conjectures," says Ono.

Accompanying this result was another achievement developing an explicit finite formula for the partition function. Previous expressions involved an infinite sum, where each term could only be expressed as an infinite non-repeating decimal number.

Counting the number of ways that a number can be 'partitioned' has captured the imagination of mathematicians for centuries. Euler, in the 1700s, was the first to make tangible progress in understanding the partition function by writing down the generating series for the function. These new results involve techniques which could have applications to other problems in number theory.


Update 1/21/11: Frank Calegari has provided a shorter proof of the result of Folsom-Kent-Ono.

This topic is also discussed on MathOverflow, with an answer by Matt Emerton.

For more background, see the Emory eScienceCommons blog

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