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On the number of solutions of the congruence \(xy\equiv l\pmod q\) under the graph of a twice continuously differentiable function. (English. Russian original) Zbl 1206.11040

St. Petersbg. Math. J. 20, No. 5, 813-836 (2009); translation from Algebra Anal. 20, No. 5, 186-216 (2008).
Summary: A result by V. A. Bykovskiĭ [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 112, 5–25 (1981; Zbl 0483.10045)] 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, J. W. Porter’s result [Mathematika 22, 20–28 (1975; Zbl 0316.10019)] on the mean number of steps in the Euclidean algorithm is sharpened and extended to the case of Gauss-Kuzmin statistics.

MSC:

11D79 Congruences in many variables
11L05 Gauss and Kloosterman sums; generalizations
11L07 Estimates on exponential sums
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11A25 Arithmetic functions; related numbers; inversion formulas
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References:

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