Given n, the set of integers m coprime to n has nonzero asymptotic density, thus so does the set mn. But then phi(n)/n > phi(mn)/mn , so the lim inf of phi(n)/n will remain the same off of a set of zero asymptotic density. So there is no such f() for asymptotic density. (I assume you mean + epsilon in your formulation.)
The Masked Avenger
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