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., Никитин Я. Ю.

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.

Ключевые слова: вероятность, probability, независимость, independence, плотность распределения, randomness, мера