Re: [math-fun] 5000 billion digits of Pi
Okay, so a natural question is whether the CF quotients have the "expected" distribution. In the invariant measure* on (0,1], the number N has probability P(N) = log_2( 1 + 1/(N(N+2)) ) So how does this compare with the calculation for pi ? --Dan * I.e., with density d(x) = (1/log(2)) 1/(x+1), invariant under the map T(x) = 1/x - [1/x]. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Fri, August 6, 2010 10:54:01 AM Subject: Re: [math-fun] 5000 billion digits of Pi Okay, so a natural question is whether the CF quotients have the "expected" distribution. In the invariant measure* on (0,1], the number N has probability P(N) = log_2( 1 + 1/(N(N+2)) ) So how does this compare with the calculation for pi ? --Dan * I.e., with density d(x) = (1/log(2)) 1/(x+1), invariant under the map T(x) = 1/x - [1/x]. _____________________________________________________________________ Correction: P(n) = (1/log 2) log((n+1)^2/n(n+2)). But also, if x is distributed according to this measure, and [1/x] = n, then y = 1/x - n has density d(y|n) = ( log((n+1)^2/n(n+2)) (n+y)(n+1+y) )^-1. So successive partial quotients are correlated. -- Gene
Dan Asimov wrote:
so a natural question is whether the CF quotients have the "expected" distribution...
Bill Gosper wrote (pi cf record):
Has anyone listed the largest terms in Hans's data? The GM?
I just figured out GM = geometric mean. :) Without getting into actual number counts, all is as expected. I calculated the geometric mean of my 180 million terms at the end of October 2002. In Mathematica, using 500-digit accuracy for the multiplications, the GM came to Khinchin + 0.00023431342958765. It should be noted that the GM-drift above and below Khinchin's constant is erratic and the closest we have come to it was much, much earlier - for 4497058 terms of the (fractional part of the) continued fraction. I have a graph of the evolving GM in this neighborhood here: http://chesswanks.com/pxp/Images/4497058.gif See also Sloane's A059101, a sequence I imagine to be infinite: http://www.research.att.com/~njas/sequences/A059101
DA> Has anyone listed the largest terms in Hans's data? The GM? If you take his file and pipe it into: tr -d \\r\ <a001203 |tr , \\n >a001203.nsv you get a file with one quotient per line. Call it a nsv file (newline separated file. :) That makes sort(1)ing, grep(1)ing and uniq(1)ing easier. The largest 36 terms are: 6175546 6523910 6666626 6875629 6894165 7151340 7293561 7702376 7733090 7794338 8093211 8372952 8646780 9286783 11095946 11866001 11948256 11986689 12051034 12354209 12996958 13165108 13228797 13297548 13435395 13743490 15880074 18130857 19626118 24183800 25787078 29491523 31296621 125962796 317579569 878783625 There are 28555 distinct terms. Except for a few pairs starting with 110 and 111, the frequency of n is greater than that of n+1, for n < 202 or so. After that there are a number of out order terms. -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
participants (4)
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Dan Asimov -
Eugene Salamin -
Hans Havermann -
James Cloos