TITLE: Inequality $\varphi(N)>\pi(N)$?
QUESTION [8 upvotes]: Is it trivial that $\varphi(N)>\pi(N)$ for sufficiently big integers $N$, where $\varphi$ is Euler's totient function and $\pi$ is the prime-counting function?
The only exceptions less than $1.000.000$ are $1,2,3,4,6,8,10,12,14,18,20,24,30,42,60,90$.
Why should the number of to $N$ relative prime numbers less than $N$ be greater than the number of primes less than $N$?
Can someone please hint on this?

REPLY [3 votes]: Hint:-
Use $\varphi(n) >\dfrac{n}{e^\gamma \ln (\ln n)+\dfrac{3}{\ln (\ln n)}}$ and $\dfrac{n}{\ln n -(1+\epsilon)}>\pi(n)$ for all sufficiently large $n$ and for all $\epsilon>0$.