Short version: I am wondering if there are any good bounds of the form $\phi(n) \geq f(n)\cdot n$ with $f(n)$ close to 1 for high $n$, optionally under the assumption that $n$ is a Carmichael number.
Long version: Consider Fermat's test in the following form for an odd integer $n>2$:
choose $a$ from $\{2, \ldots, n - 1\}$ uniformly at random
return ‘Prime’ if $[a^{n-1} = 1]\ (\text{mod}\ n)$ and ‘Composite’ otherwise
For a Carmichael number, this test incorrectly returns ‘Prime’ if and only if $a$ and $n$ are coprime, i.e. with probability $\frac{\varphi(n)-1}{n - 2}$. In textbooks, it is suggested that this number is far too high to use Fermat's test as a prime test for a Carmichael number, but I was wondering if there are any good concrete bounds for this probability that support this claim.