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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
Абзалов Л. Ф., Гатин М., Мустакимов И. А. et al., Материалы по археологии и истории античного и средневекового Причерноморья 2026 № 22 С. 395–406
The article is devoted to the analysis of the office of mushrif in the states of traditional Orient. The sources and researches characterize this institution in different ways. One authors consider mushrif as a representative of a treasure office, others states that he was a kind of ‘undercover agent” who fulfilled ruler’s secret orders in ...
Added: October 8, 2026
Kupavskii A., Noskov F., Forum of Mathematics, Sigma 2026 Vol. 14 Article 124
We call a family of $s$ sets $\{F_1, \ldots, F_s\}$ a sunflower with $s$ petals if, for any distinct $i, j \in [s]$, one has $F_i \cap F_j = \cap_{u = 1}^s F_u$. The set $C = \cap_{u = 1}^s F_u$ is called the {\it core} of the sunflower. It is a classical result of ...
Added: October 8, 2026
Popov A., Herald of the Russian Academy of Sciences 2025 Vol. 95 No. 6 P. 689–699
The article examines the export of Soviet oil to the CMEA countries in 1965–1985, relying on statistical data from declassified collections of Foreign Trade of CMEA Member Countries, which are being made available to scholars for the first time. The author argues that economic motives played a significant role in the expansion of Soviet oil ...
Added: October 8, 2026
Flamarion M. V., Pelinovsky E., Chaos, Solitons and Fractals 2026 Vol. 213 No. 2 Article 119245
This article concerns the study of modulational instability in the rotated-modified Gardner–Whitham (rmGW) equation. This model incorporates both quadratic and cubic nonlinearities, similarly to the Gardner equation, while also retaining the fully dispersive character of the Whitham equation together with a large-scale dispersive term analogous to that in the Ostrovsky equation. Using a classical multiple-scale asymptotic expansion, we ...
Added: October 8, 2026
Rezvykh P. V., Вестник Российского университета дружбы народов. Серия: Философия 2026 Т. 30 № 1 С. 127–143
The issue delivers an analysis of schellingian motives in the later works by A.F. Losev. The author brings information about Losev’s retrospective evaluation of Schelling’s influence on the formation of his own philosophical approach, it’s specific character and it’s grade. The author shows that all the main investigation projects of the later Losev which have ...
Added: October 7, 2026
Люксембург А. А., УРСС, 2005.
Изучается возможность автоматизированного построения математических теорий. Рассматривается дедуктивная система, основанная на языке логики предикатов первого порядка, объектами системы являются математические выражения или формулы, которые описывают математические объекты или их свойства. В дедуктивной системе выводятся математические определения и теоремы. Для доказательства теорем используются методы автоматического доказательства. Разработан алгоритм, выводящий часть формул системы. Для решения задачи используется аппарат математической ...
Added: October 7, 2026
Khlevniuk O. V., Journal of Slavic Military Studies 2026 Vol. 39 No. 3 P. 447–466
The Gulag as a system of terror and forced labor was the
cornerstone of Soviet mobilization efforts in the 1930s.
Paradoxically, during the war with Nazi Germany, the
Gulag’s capacities and economic significance did not grow,
but in fact declined, contrary to what one would expect. The
massive use of prisoners as a source for replenishing the
army and the huge ...
Added: October 7, 2026
Bernardin C., Gonçalves P., Olla S., Mathematical Physics Analysis and Geometry 2024 Vol. 27 No. 7
We consider the macroscopic limit for the space-time density fluctuations in the open symmetric simple exclusion in the quasi-static scaling limit. We prove that the distribution of these fluctuations converge to a gaussian space-time field that is delta correlated in time but with long-range correlations in space. ...
Added: October 6, 2026
Bernardin C., Chhaibi R., Najnudel J. et al., Probability Theory and Related Fields 2026 Vol. 195 P. 1823–1875
We study the celebrated Shiryaev-Wonham filter (Wonham, W.M., in J. Soc. Ind. Appl. Math. 347–369, 1964) in its historical setup, where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of ...
Added: October 5, 2026
Ismailov A., Spiridonov V., Успехи математических наук 2026 Т. 81 № 5 С. 183–184
Получена новая формула для цепной дроби Аски–Вильсона в форме отношения двух q-гипер-геометрических рядов. ...
Added: October 5, 2026
Selin A. A., Яблокова И. В., Вспомогательные исторические дисциплины 2026 Т. 45 № 3 С. 609–619
This article examines the possibility of reconstructing the archives of the Novgorod Prikaznaya Izba (office) from the late 16th and early 17th centuries, the materials of which were partially destroyed and partially scattered across various archival repositories. Some of these materials have been introduced into scholarly circulation, while others remain unidentified. There is a debate ...
Added: October 5, 2026
Popov A., Pivovarov N. Y., Russian Review 2026 Vol. 85 No. 4 P. 461–477
This article challenges the conventional narrative that the transferable ruble was conceived as a Soviet means of challenging Western financial hegemony. Drawing on previously unpublished archival mate rial from Russian state archives, it argues that the currency emerged primarily from a pragmatic need to settle accumulated debts within the Council for Mutual Economic Assistance (CMEA). ...
Added: October 5, 2026
[б.и.], 2026.
This collection presents papers from the 17th All-Russian Scientific Conference (2026), attended by staff and students from Russian higher education institutions, school teachers, and professionals from libraries and archives engaged in research on national history and historiography. ...
Added: October 5, 2026
Abdulkhaev K., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 P. 1–21
In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension n = 7. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces ...
Added: October 4, 2026
Вебер М., Социологическое обозрение 2026 Т. 25 № 3 С. 318–353
In the concluding part of the Second Anti-Critique (1910), Max Weber summarizes the substantive
findings of his own contributions to the debate with the historian Felix Rachfahl. In doing so, he uses the response to his relentless critic as an opportunity to articulate once again, in an explicit manner and for both his own benefit and that of the ...
Added: October 4, 2026
Kamenskii A. B., Екатеринбург: Издательство Уральского университета, 2026.
This volume has been prepared to mark the anniversary of the distinguished Ural scholar Dmitry Alekseyevich Redin. It presents research and published source texts contributed by Dmitry Alekseyevich’s colleagues, friends, and students. The diverse range of studies—covering various aspects of Russian social history from the late 16th through the 18th century, as well as issues ...
Added: October 3, 2026
Levashev V., / Series arXiv "math". 2026. No. 2609.06010.
We prove that continuous A-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity. ...
Added: September 24, 2026
Pyatov P. N., Pivovarov P. A., / Series math "arxiv.org". 2026. No. 2609.06274.
We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter.
By analyzing the ...
Added: September 24, 2026
Medvedev V., / Series arXiv "math". 2026.
We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit ...
Added: September 19, 2026
Contreras Rodríguez L., Madarro-Capó E. J., Legón-Pérez C. M. et al., Entropy 2021 Vol. 23 No. 5 Article 561
Added: September 18, 2026
Poddiakov A., / Series Social Science Research Network "Social Science Research Network". 2026. No. 7437658.
Clarity of knowledge and reasoning is necessary in many cases. Yet vagueness in scientific thinking related to surprise, curiosity, "ability to engage with not-knowing" (de Freitas) and abductive reasoning is also a crucially important source of scientific creativity which supplements combinatorial logic when dealing with the already known. Starting from studies by C. S. Peirce ...
Added: September 15, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
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
study of geodesic flows in negative curvature surface by Anosov and Sinai.
Not every closed manifold admits an Anosov flow for well-known reasons:
the fundamental group of a 3-manifold M admitting an Anosov flow must
have exponential growth, and M must be universally covered by R3. Nevertheless,
there ...
Added: August 31, 2026
Sergei L. Semakov, , in: Proceedings of the 2026 12th International Conference on Control, Decision and Information Technologies (CoDIT) (Italy, Bari, July 13–16, 2026).: IEEE, 2026. P. 153–156.
The problem of the first achievement of a given level by a random process is considered. The previously obtained general results are specified for a smooth Gaussian process. ...
Added: August 24, 2026