The probability of two integers to be co-prime, revisited -- on the behavior of CLT-scaling limit
type: Item
instance of: scholarly article
title: The probability of two integers to be co-prime, revisited -- on the behavior of CLT-scaling limit
zbMATH Open document ID: 1068.11050
author: Hiroshi Sugita
author: Satoshi Takanobu
published in: Osaka Journal of Mathematics
publication date: 2004-03-01T00:00:00Z
review text: 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 \(N\to\infty\).
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 2048565
zbMATH Keywords: greatest common divisor
zbMATH Keywords: central limit theorem
zbMATH Keywords: finite integral adèles
zbMATH Keywords: Dirichlet's theorem
reviewed by: Lothar Heinrich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2048565