# An improved inequality for the deficiency function when $\gcd(x,y)=1$, $x > 1$, and $y > 1$

(The following is an attempt to improve on the result contained in this MSE question.)

Let $$\sigma(x)$$ be the sum of the divisors of a (positive) integer $$x$$. (For example, $$\sigma(2) = 1 + 2 = 3$$.)

Define the deficiency function $$D(x)$$ to be the number $$D(x) = 2x - \sigma(x).$$

Let $$y$$ be a (positive) integer. Now I compute the difference: $$D(xy) - D(x)D(y) = 2xy - \sigma(xy) - (2x - \sigma(x))(2y - \sigma(y))$$ $$=2xy - \sigma(xy) - 4xy + 2y\sigma(x) + 2x\sigma(y) - \sigma(x)\sigma(y)$$ $$=-2xy - 2\sigma(x)\sigma(y) + 2y\sigma(x) + 2x\sigma(y) + (\sigma(x)\sigma(y) - \sigma(xy))$$ $$=2(x - \sigma(x))(\sigma(y) - y) + (\sigma(x)\sigma(y) - \sigma(xy)).$$

Now, assuming that $$\gcd(x,y)=1$$, the second term vanishes, and we are left with $$D(x)D(y) - D(xy) = 2(\sigma(x) - x)(\sigma(y) - y).$$ If we further assume that $$x>1$$ and $$y>1$$ both hold, then we have $$\sigma(x) - x \geq 1$$ $$\sigma(y) - y \geq 1$$ from which it follows that $$D(x)D(y) - D(xy) \geq 2\times{1}\times{1} = 2.$$ We therefore conclude that $$D(x)D(y) \geq D(xy) + 2.$$

Here are my questions:

QUESTIONS

(1) Is the derivation of the improved inequality correct?

(2) Will it be possible to derive a better bound than the improved inequality? Or is this already best-possible?

• (1) It looks correct to me. (2) You cannot get a better inequality than $D(x)D(y) - D(xy) \ge 2$ since the equality of this inequality holds when $a,b$ are distinct primes. (If you have some other conditions on $x,y$, then please write them in the question.) Mar 30, 2020 at 6:46
• Thank you for your time and attention, @mathlove! I know of no other conditions for $x$ and $y$. Kindly write out your last comment as an actual answer so that I may be able to accept it. Thank you! =) Mar 30, 2020 at 6:57
• I've just converted my comment into an answer. Mar 30, 2020 at 6:59

## 1 Answer

(1) It looks correct to me.

(2) You cannot get a better inequality than $$D(x)D(y) - D(xy) \ge 2$$ since the equality of this inequality holds when $$a,b$$ are distinct primes.