Monotone non-decreasing sequences of the Euler totient function

Terence Tao

Published 2023-09-05, updated 2023-09-10Version 2

Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $\varphi(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) \pi(x)$ for all $x \geq 10$, answering questions of Erd\H{o}s and Pollack--Pomerance--Trevi\~no.