# Visualizing the l-adic fractal in the partition function p(n)

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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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