# The asymptotic behavior of $\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$

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 $$\sum_{n=1}^x \frac{\sigma_a(n)\sigma_b(n)}{n^m}= \sum_{n=1}^x\bigg(\frac{\sigma_1(n)}{n}\bigg)^2 \approx c_2x\approx 2.8x$$

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.

• 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. – Will Jagy Oct 31 '18 at 1:52
• Superior Highly Composite Numbers – Mason Oct 31 '18 at 1:54
• 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 – Will Jagy Oct 31 '18 at 1:55
• 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) – Will Jagy Oct 31 '18 at 1:58
• Colossally Abundant Numbers – Mason Oct 31 '18 at 2:17