The probability of two integers to be co-prime, revisited -- on the behavior of CLT-scaling limit
Let \(\text{gcd}(x,y)\) denote the greatest common divisor of two integers \(x\) and \(y\). For any pair \((x, y)\) of integers put \(X(x, y)= 1\) if \(\text{gcd}(x, y)= 1\) and \(X(x, y)= 0\) otherwise. It is a classical result of probabilistic number theory proved by P. L. Dirichlet in 1849 that the arithmetic mean \[ S_N(x, y)= {1\over N^2} \sum^N_{m,n=1} X(x+ m,y+ n) \] converges to \(6/\pi^2\) as \(N\to\infty\) for any integers \(x\), \(y\). The main result of the paper is a kind of central limit theorem for the scaled function \(N(S_N(x, y)- 6/\pi^2)\) as \(N\to\infty\).
scientific article; zbMATH DE number 2048565