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    $\begingroup$ Very nice example! In retrospect it makes a lot of sense that questions about optimal sphere packing would require bounds on $\pi$, although the way $\pi$ comes into Viazovska's work seems to be highly nontrivial in this case, and not just through the obvious relation between $\pi$ and the volume of a ball. $\endgroup$
    – Terry Tao
    Commented May 13 at 14:32
  • $\begingroup$ Yeah, I'd say the appearance of $\pi$ here is from evaluation of Fourier series of modular forms on vertical lines in the upper half plane. $\endgroup$
    – Junyan Xu
    Commented May 13 at 16:27
  • $\begingroup$ Seewoo Lee just announced an algebraic proof in the 24 dimensional case as well! x.com/antimath3/status/1805046798765174904 $\endgroup$
    – Junyan Xu
    Commented Jun 24 at 3:13
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    $\begingroup$ @TerryTao I tend to view the appearance of 𝜋 in the formulas for the optimal density of sphere packing as an artifact of the way that human mathematicians have decided to formulate the question. As you said, the 𝜋 just comes from the formula for the volume of a unit ball. It may be more natural to measure the "density" of a sphere packing by the number of spheres per unit volume instead of by the ratio of the volume of space the spheres occupy. When the question is formulated in that way, 𝜋 doesn't appear in the answer (in dimensions 1, 2, 3, 8 and 24 where the answer is known). $\endgroup$
    – Dan Romik
    Commented Jul 17 at 21:34
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    $\begingroup$ (This reminds me of the old joke about the probability professor who, faced with a student who asked why 𝜋 appears in the formula for the limiting distribution of the number of successes in a sequence of coin tosses, replied "well, coins are round!") $\endgroup$
    – Dan Romik
    Commented Jul 17 at 21:42