In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk,

$$\frac3{(2\pi)^3}\def\intp{\int_{-\pi}^\pi}\intp\intp\intp\frac{\mathrm dx\,\mathrm dy\,\mathrm dz}{3-\cos x-\cos y-\cos z}=\frac{\sqrt6}{32\pi^3}\def\gam#1{\Gamma\left(\frac{#1}{24}\right)}\gam1\gam5\gam7\gam{11}\;,$$

contains the numbers $5$, $7$, $11$ and $\frac1{24}$ that feature prominently in the theory of the partition function, where Ramanujan's claim that the partition function does not satisfy "equally simple properties" for any primes other than $5$, $7$ and $11$ has recently been formalized and proved.

This is probably just a coincidence, but it looks intriguing – does anyone see a potential connection?

  • 5
    $\begingroup$ I doubt $5, 7, 11$ are appearing here in their role as primes, but rather in their role as the smallest residues greater than $1$ relatively prime to $24$. $\endgroup$ – Qiaochu Yuan Dec 26 '11 at 1:19
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    $\begingroup$ The whole list is not 5, 7, 11, but rather 1, 5, 7, 11. Those are all the integers coprime to 24 that are less than half of 24. The ones that are more than half of 24 are the complements of those, where the "complement" of $x$ is $24-x$. $\endgroup$ – Michael Hardy Dec 26 '11 at 17:07

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