Let $a(n)=\sigma(n)/n$ be the abundancy index of $n$ and let $F(x)$ be the distribution function of this index: i.e., the proportion of integers $n$ with $a(n)\leq x$. (This function is well-defined and continuous for the abundancy index, though that's non-trivial). Erdős proved an upper bound of the form $F(x+\frac1t)-F(x)<C/\log(t)$ and showed that this is best possible, and also proved that $F(1+\frac1t)-F(1) \in (1+o(1))e^{-\gamma}(\log t)^{-1}$, but I'm interested in the values of $F(x+\frac1t)-F(x)$ for larger $x$, and interested in inequalities the other direction: is anything known about lower bounds for $F()$ in this more general case?
(Asked a week ago on math.se with no answer)