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Inequality involving integer functions

Source: Philippine Mathematical Olympiad, 2017

December 31, 2017

inequalitiesnumber theoryfunction

Problem Statement

Given

n \in \mathbb{N}

, let

\sigma (n)

denote the sum of the divisors of

n

and

\phi (n)

denote the number of integers

n \geq m

for which

\gcd(m,n) = 1

. Show that for all

n \in \mathbb{N}

,

\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}

and determine when equality holds.

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