Friday, December 28

Pi continued fraction & Khinchin: regimes

Let the 'reduced geometric mean' be the geometric mean minus Khinchin (k). The first 25 reduced geometric means for the fractional part of π are:

 1 7-k                                              =  4.3145..
 2 105^(1/2)-k                                      =  7.5615..
 3 105^(1/3)-k                                      =  2.0322..
 4 2^(1/2)*7665^(1/4)-k                             = 10.5471.. max
 5 2^(2/5)*7665^(1/5)-k                             =  5.2088..
 6 2^(1/3)*7665^(1/6)-k                             =  2.9090..
 7 2^(2/7)*7665^(1/7)-k                             =  1.6891..
 8 2^(3/8)*7665^(1/8)-k                             =  1.2814..
 9 2^(1/3)*7665^(1/9)-k                             =  0.7182..
10 2^(3/10)*3^(1/5)*2555^(1/10)-k                   =  0.6755..
11 2^(3/11)*3^(2/11)*2555^(1/11)-k                  =  0.3248..
12 2^(1/3)*21^(1/6)*365^(1/12)-k                    =  0.7362..
13 2^(5/13)*21^(2/13)*365^(1/13)-k                  =  0.5977..
14 2^(5/14)*21^(1/7)*365^(1/14)-k                   =  0.3304..
15 2^(1/3)*21^(2/15)*365^(1/15)-k                   =  0.1164..
16 2^(3/8)*21^(1/8)*365^(1/16)-k                    =  0.0580..
17 2^(7/17)*21^(2/17)*365^(1/17)-k                  =  0.0075..
18 2^(4/9)*21^(1/9)*365^(1/18)-k                    = -0.0366..
19 2^(9/19)*21^(2/19)*365^(1/19)-k                  = -0.0755..
20 2^(9/20)*21^(1/10)*365^(1/20)-k                  = -0.1977.. max
21 2^(11/21)*21^(1/7)*365^(1/21)-k                  =  0.2561.. max
22 2^(6/11)*21^(3/22)*365^(1/22)-k                  =  0.2050..
23 2^(12/23)*21^(3/23)*365^(1/23)-k                 =  0.0746..
24 2^(1/2)*21^(1/8)*365^(1/24)-k                    = -0.0396.. max
25 2^(12/25)*3^(4/25)*5^(2/25)*7^(3/25)*73^(1/25)-k =  0.1504..

Notice that 1-17 and 21-23 are positive, while 18-20 and 24 are negative. Each one of these alternating-sign regimes has a maximum (distance from k): for instance, {20, 0.1977..} for the negative 18-20 regime. I have now calculated the start, end, and maximum for the first 27087 regimes (the final one incomplete because it ends beyond 3*10^9). Some regimes are quite lengthy, such as the positive 5418849-1434927964 and 1865143624->3*10^9. The so-far maximum in that final one is {2377934394, 0.00004194392..}, meaning that the geometric mean of 976 terms is closer to Khinchin than the geometric mean of 2377934394 terms!

[Fred Lunnon was kind enough to point me to Chapter III of William Feller's An Introduction to Probability Theory and its Applications (Volume 1) as a means of understanding some of the mathematics involved in all this.]

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