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