Monotone non-decreasing sequences of the Euler totient function

TL;DR

This paper investigates the maximum size of subsets of natural numbers up to x where the Euler totient function is non-decreasing, providing an asymptotic formula that answers longstanding questions in number theory.

Contribution

It establishes an asymptotic estimate for the largest subset with non-decreasing Euler totient values, resolving open questions posed by Erdős and others.

Findings

Derived an asymptotic formula for M(x) related to π(x)

Extended results to the sum of divisors function σ(n)

Answered longstanding open questions in number theory

Abstract

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. A similar result is also obtained for the sum of divisors function $\sigma(n)$.