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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:

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

- Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
- If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
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- Does this inequality have any solutions in $\mathbb{N}$?
- Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.
- How to prove this polynomial inequality?

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.

- Can a number be equal to the sum of the squares of its prime divisors?
- How to find all $m,n$ such that $mn|m^2+n^2+1$?
- Books about the Riemann Hypothesis
- Finding integer solutions to $y^2=x^3-2$
- Discussion on even and odd perfect numbers.
- Why can't this number be written as a sum of three squares of rationals?
- Are there useful criterions whether a positive integer is the difference ot two positive cubes?
- Probability of two events
- Algebraic independence over $\overline{\mathbb Q}$ and over $\mathbb Q$
- Find a formula for all the points on the hyperbola $x^2 - y^2 = 1$? whose coordinates are rational numbers.

Will Jagy has posted an answer to a closely related MSE question here.

Per Will Jagy’s answer in the linked MSE question (and a subsequent comment by Erick Wong), we have the *conjectured* (sharp?) bounds

$$1 \leq \frac{I(x^2)}{I(x)} \leq \prod_{p}{\frac{p^2 + p + 1}{p^2 + p}} = \frac{\zeta(2)}{\zeta(3)} \approx 1.3684327776\ldots$$

I would still be interested in an (improved) lower bound for $I(x^2)/I(x)$ when $\omega(x) \geq 3$, where $\omega(x)$ is the number of distinct prime factors of $x$.

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