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    $\begingroup$ Yes, this qualifies! I had thought to add a calculation in one of my own recent papers (page 21 of arxiv.org/pdf/2309.02325) where I needed the fact that $\zeta(2) \zeta(3)/\zeta(6) = 1.94359\dots$ was less than $2$, though this expression is not purely expressible in terms of $\pi$ so this is less compelling of an example. $\endgroup$
    – Terry Tao
    Commented May 12 at 18:48
  • 7
    $\begingroup$ One could argue this falls under exception 5, since just the trivial estimate $\sum \frac{1}{n^2} < 2 \sum \frac{1}{n(n+1)}=2$ suffices, with no need to mention $\pi$. I suppose it's a matter of taste though, since 'numerical bounds for $\pi$' is equivalent to 'numerical bounds for $\sum \frac{1}{n^2}$', but at least this way one never needs to actually calculate $\pi$ or invoke the Basel problem. $\endgroup$
    – Thomas Bloom
    Commented May 13 at 15:56