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I’m not sure that’s right. The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit
by mattkrause 4y ago
I’m not sure that’s right.
The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit of \pi^{1000}.
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- mattkrause 4y agoIOW, the arXiv title might have lost the "nth" because of some formatting thing. The title inside the PDF says "A formula for nth digit of \pi and \pi^n"
- dmurray 4y agoYes, it looks like you're right. But I don't understand the reasoning - is he actually saying you can use this formula to compute pi^n to n decimal places? So it doesn't produce arbitrary digits on their own, but excellent approximations to pi^n? And this was by taking 4 terms of the zeta function infinite product expansion. 2xn digits would presumably be given by taking more terms (but quite a few, one for every prime under 100, so in some sense this method has bad convergence when you want the mth digit of pi^n and m >> n).
- mattkrause 4y agoYeah, my sense is that the author knows this is more of a neat trick than a "practical" way of calculating pi. There's a bit on page 5 about how the Chudnovsky method totally blows this approach out of the water.
- dmurray 4y agoSeems like the real breakthrough was computing large Bernoulli numbers without a very precise value for pi, and this is a relatively easy corollary for someone to spot and get a paper out of.