Geodesic
On the number of solutions of the congruence $xy\equiv l$ $(\operatorname{mod}q)$ under the graph of a~twice continuously differentiable function
Algebra i analiz, Tome 20 (2008) no. 5, pp. 186-216.

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A result by V. A.Bykovskiĭ (1981) on the number of solutions of the congruence $xy\equiv l$ $(\operatorname{mod}q)$ under the graph of a twice continuously differentiable function is refined. As an application, Porter's result (1975) on the mean number of steps in the Euclid algorithm is sharpened and extended to the case of Gauss–Kuzmin statistics.
Keywords: Euclid algorithm, Gauss–Kuzmin statistics, Kloosterman sums. 
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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. Algebra i analiz, Tome 20 (2008) no. 5, pp. 186-216. http://geodesic.mathdoc.fr/item/AA_2008_20_5_a7/

[1] Avdeeva M. O., Raspredelenie nepolnykh chastnykh v konechnykh tsepnykh drobyakh, Preprint DVO RAN, KhO IPM No 4, Dalnauka, Vladivostok, 2000

[2] Bykovskii V. A., “Asimptoticheskie svoistva tselykh tochek $(a_1,a_2)$, udovletvoryayuschikh sravneniyu $a_1a_2\equiv l({q})$”, Zap. nauch. semin. LOMI, 112, 1981, 5–25 | MR | Zbl

[3] Vinogradov I. M., Osobye varianty metoda trigonometricheskikh summ, Nauka, M., 1976 | MR

[4] Ustinov A. V., “O statisticheskikh svoistvakh konechnykh tsepnykh drobei”, Zap. nauch. semin. POMI, 322, 2005, 186–211 | MR | Zbl

[5] Ustinov A. V., “Asimptoticheskoe povedenie pervogo i vtorogo momentov dlya chisla shagov v algoritme Evklida”, Izv. RAN. Ser. matem., 72:5 (2008), 189–224 | MR | Zbl

[6] Apostol T. M., Mathematical analysis, Addison-Wesley Publ. Co., Reading, MA etc., 1974 | MR | Zbl

[7] Estermann T., “On Kloosterman's sum”, Mathematika, 8 (1961), 83–86 | MR | Zbl

[8] Graham S. W., Kolesnik G., van der Corput's method of exponential sums, London Math. Soc. Lecture Note Ser., 126, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[9] Hardy G. H., Wrighte E. M., An introduction to the theory of numbers, Clarendon Press, Oxford Univ. Press, New York, 1979 | MR | Zbl

[10] Heath-Brown D. R., “The fourth power moment of the Riemann zeta function”, Proc. London Math. Soc. (3), 38 (1979), 385–422 | DOI | MR | Zbl

[11] Heilbronn H., “On the average length of a class of finite continued fractions”, 1969 Number Theory and Analysis, Papers in Honor of Edmund Landau, Plenum, New York, 1969, 87–96 | MR

[12] Hooley C., “On the number of divisors of a quadratic polynomial”, Acta Math., 110 (1963), 97–114 | DOI | MR | Zbl

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

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