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Question

What is the asymptotic behavior of $$\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$$

Where $\sigma_k=\sum_{d|n}d^k$

More generally I am curious if we can get bounds on $$\sum_{n=1}^x\prod_{i}\sigma_{k_i}(n)$$

Tools that I already have in my toolbox

I believe that $\sum_{n=1}^x \sigma_k \approx \frac{\zeta(k+1)}{k+1}x^{k+1}$

I have already encountered Abel's Summation which gives us a lot of power to make claims about the asymptotic behavior of sums.

Motivations: I would like to learn more about what tools are in the toolbox for handling $\sum AB$ when I know something about the $\sum A$ and the $\sum B$.

Some Efforts

I am not sure how much help my initial efforts are in tackling the main question of this post but here they are nonetheless. These efforts are related but I am starting with the simpler case of $a=b=1$ and seeing if I can make progress on that. I have made an investigation into $\sum_{n=1}^x (\frac{\sigma_1(n)}{n})^m \approx c_m x$ where $c_m$ varies in $m$. The motivation for exploring the asymptotic behavior of this particular fraction can be seen here. You can also find that $c_1=\pi^2/6$ by reading that post. Taking $m=2,a=1,b=1$ we find that $$\sum_{n=1}^x \frac{\sigma_a(n)\sigma_b(n)}{n^m}= \sum_{n=1}^x\bigg(\frac{\sigma_1(n)}{n}\bigg)^2 \approx c_2x\approx 2.8x$$

enter image description here

Pictured in the graph above is $c_m$ on the y-axis and $m$ on the x-axis. I haven't quite figured out what this function is exactly. So this is a related unsolved puzzle for me.

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  • $\begingroup$ on a related note; the way to cause very large values of the summand $\sigma_a(n) \sigma_b(n)$ is to take $n= \operatorname{LCM} \{1,2,3,4,..., k-1,k \}$ for some integer $k > 0 \; . \; $ This follows from comments of Ramanujan on his superior highly composite numbers. Actually programming the SHC numbers is a mess, but the LCM is not so bad. $\endgroup$ – Will Jagy Oct 31 '18 at 1:52
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    $\begingroup$ Superior Highly Composite Numbers $\endgroup$ – Mason Oct 31 '18 at 1:54
  • $\begingroup$ I guess. Or consider in other ways. I have never done anything with the sums you discuss, but from Hardy and Wright I know that the extreme behavior of your function is quite different from its average behavior $\endgroup$ – Will Jagy Oct 31 '18 at 1:55
  • $\begingroup$ The other famous kind are the Colossally Abundant Numbers of Alaoglu and Erdos, about 1944. These were not included in the original Ramanujan article owing to shortages of paper; fairly recently, Nicolas and Robin published the missing material from Ramanujan (about 1915) $\endgroup$ – Will Jagy Oct 31 '18 at 1:58
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    $\begingroup$ Colossally Abundant Numbers $\endgroup$ – Mason Oct 31 '18 at 2:17

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