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This page http://www.aimath.org/news/partition/ and this youtube lecture http://www.youtube.com/watch?v=aj4FozCSg8g speak of a fractal in the values of the partition function p(n).

"We prove that partition numbers are 'fractal' for every prime."

The idea of fractal to me has very visual connotations, so I was wondering if there was a way to visualize the fractals of the partition function somehow. Do they make a pretty 2d-image if formatted cleverly?

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2 Answers 2

I wouldn't take the term "fractal" too seriously (or at least too visually).

Basically, they prove that the generating function of $p(n)$ (which happens to be a modular form) has nice congruence properties modulo powers of $p$ when hit with the $U_{p^2}$ operator repeatedly. This latter operator has the effect $\sum a_nq^n\mapsto \sum a_{p^2n}q^n$ on the generating series, and thus can be thought of loosely as "$p$-adically zooming in" on the expansion. Hence the $p$-adically fractal turn of phrase.

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@Ramsey: Did you mean that $\mapsto \sum a_{p^2n}q^n$?, i.e., $q^n$ rather than $a^n$? –  Joseph O'Rourke Sep 2 '12 at 0:21
    
@Joseph O'Rourke: Indeed! Thanks for the note. –  Ramsey Sep 2 '12 at 1:57

Well, the partition function is defined for positive integers, thus discrete and inherently one-dimensional. These properties makes it harder to create "nice fractal pictures" from this function.

However, the Collatz function, (multiply by 3 and add 1 if odd, divide by 2 if even), also have these properties. The Collatz function may however be analytically extended to an entire function, and iterating this function as when constructing a Julia set, DO create pictures one would expect.

99% of all fractals I've encountered are obtained by recursion somehow, so you might need to find such recursion somehow in the computation for the theta function. If this is possible, then you might be able to draw nice pictures.

I haven't watched the movie, but there seems to be some sequence which eventually becomes a multiple of 5. Now, this sounds like escape-time calculations to me, so if this can be extended in a continuous (preferably analytic) manner, then you might have a fractal.

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