## 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].

### MSC:

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

### Keywords:

Euler’s phi-function