The distribution functions of σ(n)/n and n/φ(n), II

Let σ(n) be the sum of the positive divisors of n, and let A(t) be the natural density of the set of positive integers n satisfying σ(n)/n⩾t. We give an improved asymptotic result for logA(t) as t grows unbounded. The same result holds if σ(n)/n is replaced by n/φ(n), where φ(n) is Eulerʼs totient function.