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I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know):

Are there (known) lower and upper bounds to the following arithmetic / number-theoretic expression:

$$\frac{I(x^2)}{I(x)} = \frac{\frac{\sigma_1(x^2)}{x^2}}{\frac{\sigma_1(x)}{x}}$$

where $x \in \mathbb{N}$, $\sigma_1(x)$ is the sum of the divisors of $x$ and $I(x) = \frac{\sigma_1(x)}{x}$ is the abundancy index of $x$?

(Note that a trivial lower bound is $1$ since $x \mid x^2$ implies $I(x) \leq I(x^2)$.)

I would highly appreciate it if somebody will be able to point me to relevant references in the existing literature.

share|improve this question
I just realized that $$\displaystyle\frac{I(x^2)}{I(x)} < I(x).$$ –  Jose Arnaldo Dris Mar 5 '13 at 14:23
Consequently, a conclusive answer to this question would depend on the Riemann Hypothesis (RH), per Robin's original formulation of a condition involving the sum-of-divisors function, and which Robin showed to be equivalent to RH. –  Jose Arnaldo Dris Mar 5 '13 at 14:26

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