?
О среднем числе шагов в алгоритме Евклида с выбором минимального по модулю остатка
Математические заметки. 2009. Т. 85. № 1. С. 153–156.
Ustinov A.
The article proves an asymptotic formula for the average number of steps in the Euclidean algorithm with centered division.
Ustinov A., Известия РАН. Серия математическая 2008 Т. 72 № 5 С. 189–224
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≤c≤d≤R and R→∞, where s(c,d)=s(c/d) is the number of steps in the Euclidean algorithm applied to the numbers c and d. ...
Added: October 11, 2025
Ustinov A., Алгебра и анализ 2008 Т. 20 № 5 С. 186–216
A result by V. A.Bykovskiĭ (1981) on the number of solutions of the congruence xy≡l (modq) 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. ...
Added: October 11, 2025
Ustinov A., Математические заметки 2010 Т. 88 № 4 С. 594–604
The length of the continued-fraction expansion of a rational number with odd incomplete quotients is expressed via the Gauss–Kuzmin statistics for the classical continued fraction. This has made it possible to prove asymptotic formulas, similar to those already known for the classical Euclidean algorithm, for the mean length of the Euclidean algorithm with odd incomplete ...
Added: October 9, 2025
Ustinov A., Дальневосточный математический журнал 2011 Т. 11 № 1 С. 93–98
The article is devoted to investigation of Gauss — Kuz'min statistics for rational numbers a/b, where b is fixed, 1⩽a⩽b, (a,b)=1. New asymptotic formula for the mean value of Gauss — Kuz'min statistics is proved. It sharpens previous result which is similar to the Porter's theorem. ...
Added: October 9, 2025
Paramonova I., М.: МЦНМО, 2017.
The book contains a transcript of lecture notes of the first part of the course on Algebra read at the training courses for math teachers. Topics close to the school program in algebra are discussed. The author’s goal was to show how the school algebra can be included in a more general context of algebra as a ...
Added: December 7, 2018
Frolenkov D., Быковский В. А., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 465 № 2 С. 137–140
Получена асимптотическая формула для свертки обобщенных функций числа делителей. В качестве следствия получено улучшение результата Портера в степенной шкале для среднего числа шагов в алгоритме Евклида. ...
Added: August 27, 2016
Frolenkov D., Математический сборник 2012 Т. 203 № 2 С. 143–160
В работе рассматриваются первые моменты для числа шагов в различных алгоритмах Евклида. Для них, используя улучшенные оценки сумм дробных долей и идеи из элементарного доказательства А.Сельберга асимптотического закона распределения простых чисел, получены асимптотические формулы с новыми остаточными членами ...
Added: November 3, 2014