I botched up a proof, but I salvaged useful bits and bobs from it.
We prove by recalling the Dirichlet series of , the Euler totient function, that is,
Now we know that, for all ,
Personally, I think it is interesting, as it invokes a Dirichlet series in a somewhat unexpected way. Also, this implies that is monotonically increasing.
Challenge: Prove , using the above result or by the trivial inequality.