Robin inequality for n/phi(n)

Abstract

Let $\varphi(n)$ be the Euler function, $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function and $\gamma=0.577\ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $\sigma(n)/n < e^\gamma \log\log n$ holds for $n > 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/\varphi(n)$.