7
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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?

  • 1
    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 $24-x$.2011-12-26

0 Answers 0