While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik found the following surprising approximate identity: $$\ln{8\pi}\approx \pi\left[ 2\left(\frac{2}{3}\right)^{2/3}-\frac{1}{2}\right].$$ It is surprising because the relative accuracy is about $1.6\cdot 10^{-8}$ and I think it deserves a separate question: is this result just an isolated accident or something more profound is lurking behind it?
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asked Sep 24, 2015 at 12:44
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$\begingroup$ As a point of comparison, the closest approximation that the Inverse Symbolic Calculator gives for $\ln 8\pi$ is $(\sqrt[4]{7})(\sqrt[3]{3})(\sqrt{17})/3$, which is slightly worse than the right-hand side above. The ISC also suggests 70000/21711 as a good approximation. $\endgroup$– Timothy ChowCommented Sep 24, 2015 at 17:31
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