On the greatest common divisor of a number and its sum of divisors
Let \(\sigma(n)\) be the sum of divisors of \(n\). For real numbers \(x\geq 1\) and \(A\geq 1\), put \[ G(x,A)=\#\{n\leq x: \gcd(n,\sigma(n))>A\}. \] In the paper under review, the author investigates the behavior of \(G(x,A)\) in various ranges of \(A\) versus \(x\). For example, it is shown that if \(\beta>0\) is fixed, \(x>x_0(\beta)\) is sufficiently large and \(A\geq \exp((\log\log x)^{\beta})\), then \(G(x,A)<x/A^c\), where \(c=c(\beta)>0\). On the other hand, if \(\beta(x)\) is some function tending to zero with \(x\) and \(\varepsilon>0\) is fixed, then provided that \(x>x_0(\varepsilon,\beta)\) and \(2\leq A\leq \exp((\log\log x)^{\beta})\), the estimate \(G(A,x)\geq x/A^{\varepsilon}\) holds. It is also shown that the estimate \[ \sum_{n\leq x} \gcd(n,\sigma(m))\leq x^{1+c_1/{\sqrt{\log\log x}}} \] holds with some positive constant \(c_1\) for all \(x\geq 3\). Furthermore, for all fixed \(\varepsilon>0\), the estimate \(G(x,A)\geq x/A^{1+o(1)}\) holds uniformly for \(2\leq A\leq A^{1-\varepsilon}\). In particular, \(G(x,x^{\delta})=x^{1-\delta+o(1)}\) as \(x\to\infty\), whenever \(\delta\in (0,1)\) is fixed. Some of these results were claimed by Erdős who did not supply proofs. The proofs use an assortment of results from the anatomy of integers such as the distribution of smooth numbers and sieve methods.
scientific article; zbMATH DE number 5899031
scientific article