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Mat. Sb., 2007, Volume 198, Number 6, Pages 139–158 (Mi msb2345)  

This article is cited in 4 scientific papers (total in 4 papers)

Calculation of the variance in a problem in the theory of continued fractions

A. V. Ustinov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: We study the random variable $N(\alpha,R)=#\{j\ge1:Q_j(\alpha)\le R\}$, where $\alpha\in[0;1)$ and $P_j(\alpha)/Q_j(\alpha)$ is the $j$th convergent of the continued fraction expansion of the number $\alpha=[0;t_1,t_2,…]$. For the mean value
$$ N(R)=\int_0^1N(\alpha,R) d\alpha $$
and variance
$$ D(R)=\int_0^1(N(\alpha,R)-N(R))^2 d\alpha $$
of the random variable $N(\alpha,R)$, we prove the asymptotic formulae with two significant terms
$$ N(R)=N_1\log R+N_0+O(R^{-1+\varepsilon}), \quad D(R)=D_1\log R+D_0+O(R^{-1/3+\varepsilon}). $$

Bibliography: 13 titles.

DOI: https://doi.org/10.4213/sm2345

Full text: PDF file (548 kB)
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English version:
Sbornik: Mathematics, 2007, 198:6, 887–907

Bibliographic databases:

UDC: 511.336
MSC: Primary 11K50; Secondary 11A55
Received: 01.08.2006

Citation: A. V. Ustinov, “Calculation of the variance in a problem in the theory of continued fractions”, Mat. Sb., 198:6 (2007), 139–158; Sb. Math., 198:6 (2007), 887–907

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Ustinov, “Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm”, Izv. Math., 72:5 (2008), 1023–1059  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. V. Ustinov, “The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments”, Sb. Math., 200:4 (2009), 597–627  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. D. Frolenkov, “Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms”, Sb. Math., 203:2 (2012), 288–305  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Shparlinski I.E., “Modular hyperbolas”, Jap. J. Math., 7:2 (2012), 235–294  crossref  mathscinet  zmath  isi  elib  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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