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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On the number of solutions of the congruence $ xy\equiv l\pmod{q}$ under the graph of a twice continuously differentiable function


Author: A. V. Ustinov
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 20 (2008), nomer 5.
Journal: St. Petersburg Math. J. 20 (2009), 813-836
MSC (2000): Primary 11L05, 11L07
Published electronically: July 21, 2009
MathSciNet review: 2492364
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Abstract: A result by V. A. Bykovskiĭ (1981) on the number of solutions of the congruence $ xy\equiv l\pmod{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 Euclidean algorithm is sharpened and extended to the case of Gauss-Kuzmin statistics.


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Additional Information

A. V. Ustinov
Affiliation: Khabarovsk Division, Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, 54 Dzerzhinskiĭ Street, 680000 Khabarovsk, Russia
Email: ustinov@iam.khv.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01074-7
PII: S 1061-0022(09)01074-7
Keywords: Euclid algorithm, Gauss--Kuzmin statistics, Kloosterman sums
Received by editor(s): December 12, 2007
Published electronically: July 21, 2009
Additional Notes: Supported by RFBR (grant no. 07-01-00306), by the Far Eastern Department of the Russian Academy of Sciences (project no. 06-III-C-01-017), and by the Foundation of Assistance to the Russian Science.
Article copyright: © Copyright 2009 American Mathematical Society