Some effective inequalities between usual arithmetic functions
Let \(\tau(n)\) and \(\sigma(n)\) be the number and the sum of the divisors of \(n\). Let \(\varphi\) denote the Euler function. Many effective upper bounds are derived for \(n/\varphi(n)\) in terms of \(\varphi(n)\) and for \(\sigma(n)/n\) in terms of \(\tau(n)\). For example, it is obtained that: \[ \begin{aligned} &\frac{n}{\varphi(n)}\leq \text{e}^{\gamma}\log\log\varphi(n)+3.65279\;\;\text{if}\;n\geq 3,\\ &\frac{n}{\varphi(n)}\leq 3\log\log\varphi(n)\;\;\text{if}\;n\geq 211,\\ &\frac{n}{\varphi(n)}\leq\text{e}^{\gamma}\log\log\varphi(n)+\frac{2.95504}{\log\log\varphi(n)}\;\;\text{if}\;n\geq 7,\\ &\frac{\sigma(n)}{n}\leq 2.59791\,\log\log(3\tau(n))\;\;\text{if}\;n\geq 2, \end{aligned} \] where the symbol \(\gamma\) denotes the Euler constant.
scientific article; zbMATH DE number 5635555