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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
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