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 Mat. Zametki, 2011, Volume 90, Issue 3, Pages 431–444 (Mi mz8484)

The Law of Large Numbers for the Sum of the Partial Quotients of a Rational Number with Fixed Denominator

M. G. Rukavishnikovaab

a Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences
b Far-Eastern State University of Humanities

Abstract: We obtain a nontrivial estimate of the variance of the sum of bounded partial quotients appearing in the continued-fraction expansion of a rational number with fixed denominator. As a consequence, we obtain a law of large numbers for the sum of all partial quotients.

Keywords: law of large numbers, rational number, continued fraction, partial quotient, Euler function, Möbius function, Riemann zeta function

DOI: https://doi.org/10.4213/mzm8484

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English version:
Mathematical Notes, 2011, 90:3, 418–430

Bibliographic databases:

UDC: 511.36+511.37
Revised: 04.12.2010

Citation: M. G. Rukavishnikova, “The Law of Large Numbers for the Sum of the Partial Quotients of a Rational Number with Fixed Denominator”, Mat. Zametki, 90:3 (2011), 431–444; Math. Notes, 90:3 (2011), 418–430

Citation in format AMSBIB
\Bibitem{Ruk11} \by M.~G.~Rukavishnikova \paper The Law of Large Numbers for the Sum of the Partial Quotients of a Rational Number with Fixed Denominator \jour Mat. Zametki \yr 2011 \vol 90 \issue 3 \pages 431--444 \mathnet{http://mi.mathnet.ru/mz8484} \crossref{https://doi.org/10.4213/mzm8484} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2868371} \transl \jour Math. Notes \yr 2011 \vol 90 \issue 3 \pages 418--430 \crossref{https://doi.org/10.1134/S0001434611090100} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296476500010} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80155149505} 

• http://mi.mathnet.ru/eng/mz8484
• https://doi.org/10.4213/mzm8484
• http://mi.mathnet.ru/eng/mz/v90/i3/p431

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This publication is cited in the following articles:
1. A. A. Illarionov, “O raspredelenii tselochislennykh dlin reber poliedrov Kleina”, Dalnevost. matem. zhurn., 15:2 (2015), 214–221
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