Amount of numbers that are coprime to a Mersenne number

Question

Let $M_p = 2^p-1$ be a Mersenne number, where $p$ is prime. Is it known that almost every number in the interval $[1, M_p]$ is coprime to $M_p$? That is, is it known that $$ \lim_{p \to \infty} \dfrac{\phi(M_p)}{M_p} = 1? $$ Here $\phi$ is the Euler's totient function. If so, do you know a reference? Thanks.

Answer 1

I originally intended to obtain a reference for this seemingly interesting result, but it seems it is not to be easily found on the web. Therefore I've decided to write the short proof for the convenience of other readers.

Let $\omega(k)$ denote the number of prime divisors of an integer $k$. Now by Euler's product formula for $\phi$ and since each prime factor of $M_p$ is greater than $p$, we have $$ \dfrac{\phi(M_p)}{M_p} = \prod_{\ell \mid M_p} \left( 1 - \dfrac{1}{\ell} \right) > \left( 1 - \dfrac{1}{p} \right)^{\omega(M_p)} \to 1 $$ as $p \to \infty$, since $$ \lim_{p \to \infty}\dfrac{\omega(M_p)}{p} = 0. $$ The last can be easily shown to be true from the fact that $\omega(n)$ is bounded on prime powers, while $\log_2(n)$ is not. Finally since $\phi(M_p)/M_p < 1$, the Squeeze Theorem implies the result.