Geodesic
The average length of finite continued fractions with fixed denominator
Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 644-683 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In 1968 Heilbronn proved an asymptotic formula for the mean value of the lengths of continued fraction expansions of rational numbers with identical denominators. A new method is proposed for solving Heilbronn's problem and its generalizations. New estimates for the remainders, which improve the earlier results due to Porter (1975) and Ustinov (2005), are obtained. Bibliography: 28 titles.
Keywords: continued fraction, additive divisor problem
Mots-clés : convolution formula. 
@article{SM_2017_208_5_a2,
     author = {V. A. Bykovskii and D. A. Frolenkov},
     title = {The average length of finite continued fractions with fixed denominator},
     journal = {Sbornik. Mathematics},
     pages = {644--683},
     year = {2017},
     volume = {208},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_5_a2/}
}
TY  - JOUR
AU  - V. A. Bykovskii
AU  - D. A. Frolenkov
TI  - The average length of finite continued fractions with fixed denominator
JO  - Sbornik. Mathematics
PY  - 2017
SP  - 644
EP  - 683
VL  - 208
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2017_208_5_a2/
LA  - en
ID  - SM_2017_208_5_a2
ER  - 
%0 Journal Article
%A V. A. Bykovskii
%A D. A. Frolenkov
%T The average length of finite continued fractions with fixed denominator
%J Sbornik. Mathematics
%D 2017
%P 644-683
%V 208
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2017_208_5_a2/
%G en
%F SM_2017_208_5_a2
V. A. Bykovskii; D. A. Frolenkov. The average length of finite continued fractions with fixed denominator. Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 644-683. http://geodesic.mathdoc.fr/item/SM_2017_208_5_a2/

[1] V. A. Bykovsky, N. V. Kuznetsov, A. I. Vinogradov, “Generalized summation formula for inhomogeneous convolution”, Automorphic functions and their applications (Khabarovsk, 1988), Acad. Sci. USSR Inst. Appl. Math., Khabarovsk, 1990, 18–63 | MR | Zbl

[2] T. M. Dunster, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter”, Siam J. Math. Anal., 21:4 (1990), 995–1018 | DOI | MR | Zbl

[3] T. Estermann, “On the representations of a number as the sum of two products”, Proc. London Math. Soc., S2-31:1 (1930), 123–133 | DOI | MR | Zbl

[4] H. Halberstam, “An asymptotic formula in the theory of numbers”, Trans. Amer. Math. Soc., 84:2 (1957), 338–351 | DOI | MR | Zbl

[5] H. Heilbronn, “On the average length of a class of finite continued fractions”, Abhandlungen aus Zahlentheorie und Analysis [Number theory and analysis], Zur Erinnerung an E. Landau [Papers in honor of E. Landau], VEB, Berlin; Plenum, New York, 1968, 87–96 | MR | Zbl

[6] A. E. Ingham, “Some asymptotic formulae in the theory of numbers”, J. London Math. Soc., 2:3 (1927), 202–208 | DOI | MR | Zbl

[7] D. E. Knuth, “Evaluation of Porter's constant”, Comput. Math. Appl., 2:2 (1976), 137–139 | DOI | Zbl

[8] L. Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York–Amsterdam, 1981, xvii+359 pp. | MR | Zbl

[9] L. Lewin, “The dilogarithm in algebraic fields”, J. Austral. Math. Soc. Ser. A, 33:3 (1982), 302–330 | DOI | MR | Zbl

[10] G. Lochs, “Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmäßigen Kettenbrüche”, Monatsh. Math., 65:1 (1961), 27–52 | DOI | MR | Zbl

[11] Y. Motohashi, “The binary additive divisor problem”, Ann. Sci. École Norm. Sup. (4), 27:5 (1994), 529–572 | MR | Zbl

[12] NIST handbook of mathematical functions, eds. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, Ch. W. Clark, U. S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge Univ. Press, Cambridge, 2010, xvi+951 pp. | MR | Zbl

[13] R. B. Paris, D. Kaminski, Asymptotics and Mellin–Barnes integrals, Encyclopedia Math. Appl., 85, Cambridge Univ. Press, Cambridge, 2001, xvi+422 pp. | DOI | MR | Zbl

[14] J. W. Porter, “On a theorem of Heilbronn”, Mathematika, 22:1 (1975), 20–28 | DOI | MR | Zbl

[15] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Stud. Adv. Math., 46, Cambridge Univ. Press, Cambridge, 1995, xvi+448 pp. | MR | Zbl

[16] E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1951, vi+346 pp. | MR | Zbl

[17] V. A. Bykovskii, D. A. Frolenkov, “Asymptotic formula for the convolution of a generalized divisor function”, Dokl. Math., 92:3 (2015), 670–673 | DOI | MR | Zbl

[18] V. A. Bykovskii, D. A. Frolenkov, “Asimptoticheskie formuly dlya vtorykh momentov $L$-ryadov golomorfnykh parabolicheskikh form na kriticheskoi pryamoi”, Izv. RAN. Ser. matem., 81:2 (2017), 5–34 | DOI

[19] A. A. Karatsuba, S. M. Voronin, The Riemann zeta-function, de Gruyter Exp. Math., 5, Walter de Gruyter Co., Berlin, 1992, xii+396 pp. | DOI | MR | MR | Zbl | Zbl

[20] I. S. Gradshteyn, I. M. Ryzhik, “Table of integrals, series, and products”, 5th ed., Academic Press, Inc., Boston, MA, 1994, xlviii+1204 pp. | MR | MR | Zbl | Zbl

[21] A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin, 1993, xiv+222 pp. | DOI | MR | MR | Zbl

[22] N. V. Kuznetsov, Convolution of the Fourier coefficients of the Eisenstein–Maass series, 29:2 (1985), 1131–1159 | DOI | MR | Zbl

[23] F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York–London, 1974, xvi+572 pp. | MR | MR | Zbl | Zbl

[24] A. V. Ustinov, “On the statistical properties of finite continued fractions”, J. Math. Sci. (N. Y.), 137:2 (2006), 4722–4738 | DOI | MR | Zbl

[25] A. V. Ustinov, “The mean number of steps in the Euclidean algorithm with least absolute value remainders”, Math. Notes, 85:1 (2009), 142–145 | DOI | DOI | MR | Zbl

[26] A. V. Ustinov, “The mean number of steps in the Euclidean algorithm with odd incomplete quotients”, Math. Notes, 88:4 (2010), 574–584 | DOI | DOI | MR | Zbl

[27] 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 | DOI | MR | Zbl

[28] A. V. Ustinov, “O statistikakh Gaussa–Kuzmina v korotkikh intervalakh”, Dalnevost. matem. zhurn., 11:1 (2011), 93–98 | MR | Zbl