15
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$ Sep 24, 2015 at 17:31

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.