TITLE: Typical value of totient function
QUESTION [12 upvotes]: Can anyone tell me what the expected value of Euler's totient function φ$(n)$ is (roughly) if you choose a random integer $n$ in the range $[N,N+M]$, where $M$ is large and $N$ is larger than $M$? (I think of $M$ as being $cN$ for some small constant $c$, which, if one wanted an answer accurate to $1+o(1)$, would in reality be a slowly decreasing function of $N$.)

REPLY [4 votes]: Let me also mention the following:
You can adapt Schoenberg's result to prove that
1/M * {N <= n <= N + M : phi(n) / n <= t} --> F(t) 
uniformly in t, where F is a distribution function. The proof goes
by computing the moments sum((phi(n)/n)^k , N <= n <= N + M).
You can probably get a O(loglog N / log N) rate of convergence
(as was done by Levin ... if I recall correctly).