Monotone nondecreasing sequences of the Euler totient function (Q6564023)

In this interesting and very well written article, the author proves that, for all \(x \ge 10\), we have \N\[\N\pi(x) \le M(x) \le \Biggl( 1 + O \left( \frac{(\log \log x)^5}{\log x}\right) \Biggr) \pi(x),\N\]\Nwhere \(M(x)\) denotes the largest cardinality of a subset of \(\left\lbrace n \in \mathbb{Z}_{\ge 1} : n \le x \right\rbrace\) on which the Euler totient function \(\varphi(n)\) is nondecreasing. This answers a question of \textit{P. Erdős} [Resen. Inst. Mat. Estat. Univ. São Paulo 2, No. 2, 165--186 (1995; Zbl 0871.11002)]. As the author points it out, it is surmised that, for all \(x \ge 31957\), one has \(M(x) = \pi(x) + 64\), but this conjecture seems to be currently out of reach. A key part in the proof is played by the astonishing inequality \N\[\N\sum_{\substack{d \ge 1 \\\N\frac{\varphi(d)}{d}=q}} \frac{1}{d} \le 1\N\]\Nvalid for any positive rational number \(q\), with equality for \(q=1\) or \(q = \frac{1}{2}\). In the final section of the article, the author discusses about obstacles to improvement, possible conditional results, variants with other multiplicative functions, such as the sum-of-divisor function \(\sigma\) which is `close' to the Euler totient, or some open questions.