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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)

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    $\begingroup$ Does Theorem 1 of this paper dx.doi.org/10.4064/aa121-4-6 answer your question? As far as I can tell nothing much more is known, but Paul Pollack has recent works on this topic and would maybe know if there is. $\endgroup$
    – Sary
    Apr 26, 2023 at 22:23

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