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On the coprimality of some arithmetic functions. (English) Zbl 1299.11069
From the text: Let $$\varphi$$ denote the Euler function. Given a positive integer $$n$$, let $$\sigma(n)$$ denote the sum of the positive divisors of $$n$$ and let $$\tau(n)$$ be the number of divisors of $$n$$. We obtain an asymptotic estimate for the counting function of the set $$R(x):=\#\{n\leq x:\gcd(\varphi(n),\tau(n))=\gcd(\sigma(n),\tau(n))=1\}$$. Moreover, setting $$l(n):=\gcd(\tau(n),\tau(n+1))$$, we provide an asymptotic estimate for the size of $$N(x):=\#\{n\leq x: l(n)=1\}$$. Main results:
Theorem 1: As $$x\to\infty$$, we have $$R(x)=c_1(1+o(1))\frac x{\sqrt{\log x}}$$, where $$c_1$$ is a suitable constant.
Theorem 2: As $$x\to\infty$$, we have $$N(x)=c_2(1+o(1)) \sqrt x$$ for some positive constant $$c_2$$.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11A25 Arithmetic functions; related numbers; inversion formulas
##### Keywords:
number of divisors; sum of divisors
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