On the distribution of some means concerning the Euler function (Q1038648)

A real sequence \(a_n\) is dense modulo one, if the set of fractional part of values of it becomes dense in the unit interval \([0,1]\). The aim of the paper under review is studying density of some sequences related by the well-known means of the values of Euler function \(\varphi\). The authors prove that the sequences with general terms \(a_n=\frac1n \sum_{m\leq n}\varphi(m)\), \(g_n=\left(\prod_{m\leq n}\varphi(m)\right)^{1/n}\), \(h_n=n/\sum_{m\leq n}\frac{1}{\varphi(m)}\) and \(s_n=\sqrt{na_n}\) all are dense modulo one. The key point of the proofs is considering the ratio \(\varphi(n+j)/(n+j)\) when \(j\) runs over a certain set of integers obtained by Chinese remainder theorem.