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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 5, Pages 189–224 (Mi izv2640)  

This article is cited in 11 scientific papers (total in 11 papers)

Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm

A. V. Ustinov

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

Abstract: We prove asymptotic formulae with two significant terms for the expectation and variance of the random variable $s(c/d)$ when the variables $c$ and $d$ range over the set $1\leq c\leq d\leq R$ and $R\to\infty$, where $s(c,d)=s(c/d)$ is the number of steps in the Euclidean algorithm applied to the numbers $c$ and $d$.

DOI: https://doi.org/10.4213/im2640

Full text: PDF file (642 kB)
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English version:
Izvestiya: Mathematics, 2008, 72:5, 1023–1059

Bibliographic databases:

UDC: 511.335+511.336
MSC: 11K50, 11A55
Received: 27.03.2007

Citation: A. V. Ustinov, “Asymptotic behaviour of the first and second moments for the number of steps in the Euclidean algorithm”, Izv. RAN. Ser. Mat., 72:5 (2008), 189–224; Izv. Math., 72:5 (2008), 1023–1059

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Ustinov, “On the number of solutions of the congruence $xy\equiv l$ $(\operatorname{mod}q)$ under the graph of a twice continuously differentiable function”, St. Petersburg Math. J., 20:5 (2009), 813–836  mathnet  crossref  mathscinet  zmath  isi  elib
    2. A. V. Ustinov, “The Mean Number of Steps in the Euclidean Algorithm with Least Absolute-Value Remainders”, Math. Notes, 85:1 (2009), 142–145  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. 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
    4. E. N. Zhabitskaya, “The average length of reduced regular continued fractions”, Sb. Math., 200:8 (2009), 1181–1214  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. A. V. Ustinov, “The Mean Number of Steps in the Euclidean Algorithm with Odd Incomplete Quotients”, Math. Notes, 88:4 (2010), 574–584  mathnet  crossref  crossref  mathscinet  isi
    6. E. N. Zhabitskaya, “Mean Value of Sums of Partial Quotients of Continued Fractions”, Math. Notes, 89:3 (2011), 450–454  mathnet  crossref  crossref  mathscinet  isi
    7. O. A. Gorkusha, “O srednei dline diagonalnykh drobei Minkovskogo”, Dalnevost. matem. zhurn., 11:1 (2011), 10–27  mathnet  elib
    8. 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
    9. D. Frolenkov, “The mean value of Frobenius numbers with three arguments”, Izv. Math., 76:4 (2012), 760–819  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Shparlinski I.E., “Modular hyperbolas”, Jap. J. Math., 7:2 (2012), 235–294  crossref  mathscinet  zmath  isi  elib  scopus
    11. Valérie Berthé, Hitoshi Nakada, Rie Natsui, Brigitte Vallée, “Fine costs for Euclidʼs algorithm on polynomials and Farey maps”, Advances in Applied Mathematics, 2014  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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