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Khinchin Harmonic Mean

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The Khinchin (or Khintchine) harmonic mean is associated with the simple continued fraction [a_0;a_1,a_2,...] of a "generic" real number, where a_i are the partial quotients. It is defined by

K_(-1)=lim_(n->infty)n/(a_1^(-1)+a_2^(-1)+...+a_n^(-1)).
(1)

This definition is analogous to that of Khinchin's constant K, but with the partial quotients raised to the -1 power. The mean exists and has a unique common value (except for a set of real numbers with measure zero) given by

K_(-1)=ln2[sum_(n=1)^(infty)ln[(1-1/((n+1)^2))^(-1/n)]]^(-1)
(2)
=-ln2[sum_(n=1)^(infty)1/nln(1-1/((n+1)^2))]^(-1)
(3)
=1.745405662407346863494596309...
(4)

(OEIS A087491; Bailey et al. 1997, Plouffe).

Khinchin's constant K=K_0 and the Khinchin harmonic mean K_(-1) are just two of an infinite family of such constants K_p, the first few of which are summarized in the following table.

pOEISvalue
0A0022102.685452001065306445309714835481795693820382293994462
-1A0874911.745405662407346863494596309683661067294936618777984
-2A0874921.450340328495630406052983076680697881408299979605904
-3A0874931.313507078687985766717339447072786828158129861484792
-4A0874941.236961809423730052626227244453422567420241131548937
-5A0874951.189003926465513154062363732771403397386092512639671
-6A0874961.156552374421514423152605998743410046840213070718761
-7A0874971.133323363950865794910289694908868363599098282411797
-8A0874981.115964408978716690619156419345349695769491182230400
-9A0874991.102543136670728013836093402522568351022221284149318
-10A0875001.091877041209612678276110979477638256493272651429656

See also

Khinchin's Constant

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References

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417-431, 1997.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 161, 2003.Khinchin, A. Ya. "Average Values." §16 in Continued Fractions. New York: Dover, pp. 86-94, 1997.Plouffe, S. "The Khintchine Harmonic Mean." https://www.plouffe.fr/simon/constants/khintchine1.txt.Sloane, N. J. A. Sequences A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, and A087500 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Khinchin Harmonic Mean

Cite this as:

Weisstein, Eric W. "Khinchin Harmonic Mean." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KhinchinHarmonicMean.html

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