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On the Probability of Co-primality of two Natural Numbers Chosen at Random: From Euler identity to Haar Measure on the Ring of Adeles
Bernoulli News. 2017. Vol. 24. No. 1. P. 7–13.
Abramovich S., Nikitin Y. Y.
The paper describes the rich history of remarkable problem lying at the confluence of number theory and probability. What is the probability of co-prima;lity of two randomly selected random numbers? In russian literature this problem and its solution is attributed to P.L.Chebyshev, but we show that before Chebyshev the solution was given by P. Dirichlet, who probably learned the paper from the great Gauss. We describe three main approaches to the solution of this problem and give numerous generalizations of it.
Priority areas:
mathematics
Language:
English
Chemical Papers 2026
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Glutsyuk A., / Series arXiv "math". 2026.
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Glutsyuk A., / Series arXiv "math". 2026.
A planar dual billiard is a planar curve γ equipped with a family (σP)|P∈γ of projective involutions of the projective lines LP tangent to γ at P that fix P. A dual billiard is called rationally integrable, if there exists a rational function R(x,y) of two variables (called first integral) whose restriction to each tangent ...
Added: September 8, 2026
Alexandrov A., Glutsyuk A., Journal of Differential Equations 2026 Vol. 465 Article 114178
The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on ...
Added: September 8, 2026
Glutsyuk A., Inventiones Mathematicae 2026 Vol. 245 P. 977–1058
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study ...
Added: September 8, 2026
Prikhodko Artem, Kubrak D., Compositio Mathematica 2026 Vol. 162 No. 6 P. 1377–1438
In this follow-up paper we show that smooth Hodge-proper stacks over O𝐾 are ℚ𝑝-locally acyclic: namely the natural map between étale ℚ𝑝-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the ℚ𝑝-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. ...
Added: September 7, 2026
Ismailov A., Constructive Approximation 2026
The measure of the positivity set {x ∈ [0; 2π] | f (x) > 0} of a trigonometric polynomial f is bounded from below by the Motzkin density.
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Added: September 7, 2026
Kucheryavyy P., Математические заметки 2026 Т. 2026 № 120 С. 380–401
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Added: September 7, 2026
Филянов Н. А., Вестник древней истории 2026 Т. 86 № 2 С. 388–408
Данная статья посвящена вопросу о возможном влиянии законопроекта плебейского трибуна 126 г. до н.э. М. Юния Пенна об изгнании перегринов из Рима на гракханскую политику в т.н. италийском вопросе, которая была представлена рогацией консула 125 г. до н.э. М. Фульвия Флакка о предоставлении союзникам (латинам) прав римского гражданства или права провокации (provocatio ad populum). По ...
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Voloshina S., Логос 2025 Т. 35 № 2 (165) С. 193–233
Статья посвящена исследованию маскарадов как социального и культурного феномена в эпоху правления Николая I. Автор рассматривает маскарады не только как развлекательные мероприятия, но и как сложные ритуалы, совмещающие властные сценарии, гендерные роли, социальные иерархии и индивидуальные характеры. Основное внимание уделяется трем типам маскарадов: придворным, аристократическим и публичным. Придворные маскарады (новогодние и июльские) служили инструментом демонстрации ...
Added: September 4, 2026
Voloshina S., СПб.: Владимир Даль, 2016.
Представленная книга - первый опыт "полной" биографии Николая Платоновича Огарёва, попытка объемного описания этой разносторонней, талантливой личности, его многочисленных и неоднозначных интересов и занятий, политических, поэтических, дружеских и финансовых связей с современниками (А. И. Герценом, М. А. Бакуниным, Н. А. Некрасовым и многими другими). Тесная связь биографии Н. П. Огарёва с современными ему социальными, политическими ...
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Voloshina S., М.: Издательский дом «Дело» РАНХиГС, 2022.
Книга посвящена взаимоотношениям власти и журналистики 1830–1855 гг. – времени правления Николая I и одновременно периода журналистской деятельности Андрея Александровича Краевского. Это годы профессионализации журналистики и литературы, попыток общественного мнения выразить себя в печати посредством завуалированных общественно-политических высказываний в условиях жесточайшей цензуры. Один из фокусов исследования в книге – профессиональная биография Краевского, талантливого редактора и предпринимателя, ...
Added: September 4, 2026
Потехина И. П., СПб.: Евразия, 2026.
В центре внимания этой книги развитие папской власти в период с середины VIII до конца XII в. — в эпоху, когда римская церковь из локального религиозного центра, пусть и засвидетельствовавшего свое апостольское происхождение, постепенно превращалась в одну из главных политических сил европейского континента. Автор показывает, как выстраивались отношения преемников ап. Петра с франкскими правителями, византийскими ...
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Pochinka O., Shmukler V., / Series math.RT "arXiv:1808.06395 [math.RT]". 2026.
Anosov flows have a long and rich history, firstly motivated by the
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the fundamental group of a 3-manifold M admitting an Anosov flow must
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In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers S1,S2≥1 of generalized quantum measurements at two sites. In the present article, we find analytically a new general locality condition sufficient for a nonseparable Werner state with a ...
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Bolbachan V., / Series math "arxiv.org". 2024.
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