, Volume 30, Issue 3, pp 379-398

Sets of monotonicity for Euler’s totient function

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Abstract

We study subsets of [1,x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any ϵ>0, every such subset has size smaller than ϵx, once x>x 0(ϵ). This confirms a conjecture of the second author.

P. Pollack was supported by NSF award DMS-0802970 while C. Pomerance and E. Treviño were supported by award DMS-1001180.