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Part of 2017 Philippine MO

Problems(1)

Inequality involving integer functions

Source: Philippine Mathematical Olympiad, 2017

12/31/2017
Given n∈Nn \in \mathbb{N}, let σ(n)\sigma (n) denote the sum of the divisors of nn and ϕ(n)\phi (n) denote the number of integers n≥mn \geq m for which gcd⁡(m,n)=1\gcd(m,n) = 1. Show that for all n∈Nn \in \mathbb{N},
1σ(n)+1ϕ(n)≥2n\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}
and determine when equality holds.
inequalitiesnumber theoryfunction