On the proportion of numbers coprime to a given integer. (English) Zbl 1175.11055

De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 47-64 (2008).
Let \(\varphi(n)\) denote Euler’s function, and let \(a(n)=\frac{\varphi(n)}{\gcd(n,\varphi(n))}\), \(b(n)=\frac{n}{\gcd(n,\varphi(n))}\). For positive integers \(a,b\), let \[ f(a)=\#\{n\,\text{squarefree}:a(n)=a\}, \]
\[ g(b)=\#\{n\,\text{squarefree}:b(n)=b\}. \] The authors present algorithms for computing the values \(f(a), g(b)\). They discuss the maximal orders of the arithmetic functions \(f(a)\), \(g(b)\), and they show that these functions are normally \(0\). They also discuss the arithmetical function \(\gcd(n,\varphi(n))\). They show that normally it is the largest divisor of \(n\) composed of primes at most \(\log\log n\), and on average it is bounded by \(n^{o(1)}\). This has an application in the counting of solutions of certain polynomial congruences. By this paper, the first author obtains Luca number 1.
For the entire collection see [Zbl 1142.11002].


11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions