MathDB

Let

a

be a positive real number. Prove that
a) There exists

n\in \mathbb{N}

with

\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a

.
b) There exists

n\in \mathbb{N}

with

\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a

.
(As usual,

\sigma(n) = \sum_{d\mid n} d

and

\varphi(n)

is the number of integers

1\le m\le n

which are coprime with

n

.)
Proposed by Deniz Can Karaçelebi

Problem Statement