This is an interesting problem I ran into last week. I thought I might post it as I have very little clue about how to actually solve it. The proof I turned in concluded with "lots of arm waving... QED" (which actually got a fair amount of credit since no one solved it).

Let o(n) denote the Euler totient function. That is, o(n) is the number of integers relatively prime to n that are less than n.

What is the infimum limit as n approached infinity of o(n)/n ?

I reasoned that it is 0 because if you take the k = p1*p2*...*pm (where the p's are primes to the first power), then o(k)/k = (p1 - 1)*(p2 - 1)*...*(pm - 1)/(p1*p2*...*pm), which looks somewhat like 1*2*3*4...*(n-1)/(2*3*5*...*n) which clearly goes to 0 as n approaches infinity... though most of the terms are missing, so I am not sure.

The other thought was products of n*log(n) [roughly the nth prime], though I was unable to get anywhere with this.

Anyways... an interesting problem... anyone got any ideas on it?