A Robin inequality for \(n/\varphi(n)\) (Q6546425)

Let \(\sigma(n)=\sum_{d|n}d\) be the sum of (positive) divisors function. \textit{G. Robin} [J. Math. Pures Appl., IX. Sér. 63, 187--213 (1984; Zbl 0516.10036)] showed that the following inequality, known as the Robin's inequality, is equivalent with the validity of the Riemann hypothesis,\N\[\N\frac{\sigma(n)}{n}<e^\gamma\log\log n\qquad(n>5040),\N\]\Nwhere \(\gamma\) is the Euler constant. In the paper under review, the author obtains an analogue result concerning the Euler function \(\varphi\), by proving that the inequality\N\[\N\frac{n}{\varphi(n)}<e^\gamma\log\log n+\frac{e^\gamma(4+\gamma-\log(4\pi))}{\sqrt{\log n}}\qquad(n>A),\N\]\Nwhere \(\log A=1590171.636107\ldots\), is an equivalent to the validity of the Riemann hypothesis. The proof is based on the concept of super champions of the function \(n/\varphi(n)\).