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Let $s(n)$ denote the sum of the proper divisors of $n$. For example, $s(6)=1+2+3=6$.

In 1976, Erdős claimed that (top of page 642) for every $k$ and $\delta>0$ and for all $n$ except a sequence of density $0$, it is true that $$(1-\delta)n\left(\frac{s(n)}{n}\right)^i

In 1990, Erdős retracted his claim that (middle of page 169) for every $k$ and $\epsilon>0$ and for all $n$ except a sequence of density $0$, it is true that $$\left|\frac{s(n)}{n}-\frac{s^{j+1}(n)}{s^j(n)}\right|<\epsilon\text{ for every }1\leqslant j\leqslant k.$$

I conclude that these inequalities are supposed to be equivalent, but I cannot show that this is the case. Are these inequalities equivalent?

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    Which inequalities are supposed to be equivalent? – 2017-02-23
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    @FluidizedPigeonReactor $(1-\delta)n\left(\frac{s(n)}{n}\right)^i – 2017-02-23
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    what processes did you use to conclude that they are equivalent? – 2017-03-05

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