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Abstract

Let $n$ be a positive integer, $\varphi (n)$ the Euler totient function, and $\sigma (n)=\sum _{d\mid n}d$ the sum of the divisors of $n$. It is easy to prove that $\sigma (n)/n\le n/\varphi (n)$. Landau proved that when $n\to \infty $, $\limsup n/(\varphi (n)\log \log n) = e^\gamma $, where $\gamma =0.577\ldots $ is the Euler constant, and a few years later, Gronwall proved that $\limsup \sigma (n)/(n\log \log n)$ is also equal to $e^\gamma $. Afterwards, several authors gave effective upper bounds for $n/\varphi (n)$ and $\sigma (n)/n$, either under the Riemann hypothesis or without assuming it. Let $X \ge 4$ be a real number and $\Phi (X)$ the maximum of $n/\varphi (n)$ for $n\le X$. Similarly, we denote by $\Sigma (X)$ the maximum of $\sigma (n)/n$ for $n\le X$. Our first result gives effective upper and lower bounds for $\Phi (X)/\Sigma (X)$. Next, we give new effective upper bounds for $n/\varphi (n)$ and for $\sigma (n)/n$.