TITLE: The limit superior of $\varphi(2^n-1)/(2^n-1)$
QUESTION [13 upvotes]: Let $\varphi$ denote Euler's totient function. It's widely believed that $2^n-1$ is prime for infinitely many (prime) $n$, which, in turn, implies
$$\limsup_{n \to \infty} \frac{\varphi(2^n-1)}{2^n-1} = 1.
$$
But what about an unconditional proof? This is sort of a follow-up of a question asked by @kodlu yesterday.

REPLY [29 votes]: Let $n$ be a large prime. Then $2^n-1=p_1\dots p_k$ (possibly with equal factors), where $n\mid p_i-1$ for all $i$. Therefore, $k\leq \log_n(2^n-1)<\frac{n}{\log_2n}$. Thus,
$$
  \frac{\varphi(2^n-1)}{2^n-1}\geq\prod_{i=1}^k\left(1-\frac1{p_i}\right)
  \geq \left(1-\frac1n\right)^{n/\log_2n}\to1, \quad n\to\infty,
$$
as required.