What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.
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5$\begingroup$ Chudnovsky's algorithm produces 14.18 digits of $\pi$ per iteration. $\endgroup$– Carlo BeenakkerFeb 2, 2017 at 7:15
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1$\begingroup$ @CarloBeenakker Is this 14.18 number known to have a closed form? $\endgroup$– WojowuFeb 2, 2017 at 17:53
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$\begingroup$ @Wojowu --- I think it does, see below. $\endgroup$– Carlo BeenakkerFeb 2, 2017 at 18:54
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1 Answer
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[I'm following up on my comment, in response to Wojowu's query:]
The number of digits $d$ of $1/\pi=\sum_{k=0}^\infty c_k$ produced per iteration by the Chudnovsky algorithm, which has a linear convergence, follows from $10^d=\lim_{k\rightarrow\infty}|c_{k}/c_{k+1}|$, hence $$d={}^{10}\log 151931373056000=14.1816\cdots$$
in connection with the unusual logarithm notation, I asked at HSM and got an informative response: the notation from the early 19th century for the base of the logarithm by A.L. Crelle was a superscript either in front $^{b}\!\log$ or above $\overset{b}{\log}$ --- see page 107 of A History of Mathematical Notations (volume II).
answered Feb 2, 2017 at 18:49
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1$\begingroup$ Too bad it isn't 14.15926535... Gerhard "Strong Law For Irrational Numbers?" Paseman, 2017.02.02. $\endgroup$ Feb 2, 2017 at 21:39
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1$\begingroup$ Yes, is it an unconventional notation? $\endgroup$ Feb 2, 2017 at 22:25
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7$\begingroup$ @CarloBeenakker I've personally never seen it before. $\endgroup$ Feb 3, 2017 at 3:37
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2$\begingroup$ this is curious; it may very well be a Dutch thing, at least the Dutch Wikipedia entry gives both alternative notations ${}^{a}\!\log$ and $\log_{a}$ for the base of the logarithm, and I definitely learned the former at school. $\endgroup$ Feb 3, 2017 at 7:32