I want to know how to program this formula (https://people.math.gatech.edu/~mschmidt34/images/sum-of-divisors-exact-formula.png)
but I can't understand the math behind or the several variables used to compute? From a math understanding ending at Linear Algebra how could I go about learning this.
here's the article behind it https://arxiv.org/abs/1705.03488
On
The article in question has officially been accepted for publication in the Journal of Integer Sequences (to appear, 2018). In place of Google Drive links which I have included as references to supplementary computational material in several of my recent papers, I decided to setup a more permanent home for this sort of computational data. A Mathematica notebook for this particular article is available here. It includes full implementations of the formulas in this preprint. I believe this is exactly what you were asking for, though I should point out that next time a more direct route would be to just contact the author of the work you're interested in!
I have no idea why this posted ended up getting tagged with Galois theory. First of all, make sure you have viewed the most recent version of the article to date. This newer version of the manuscript has been resubmitted for review. There were some typos in the original manuscript that I used when I posted that image on my website. The updated version of the identity is as follows: $$\sigma_{\alpha}(x) = H_x^{(1-\alpha)} + \tau_{\alpha}(x) + S_1(x) + S_2(x),$$ where $S_i(x)$ correspond to the sums over the primes in the image and $$\tau_{\alpha}(x) = H_{\lfloor x/d \rfloor}^{(1-\alpha)} d^{\alpha} c_d(x), $$ is defined in terms of Ramanujan's sum $c_d(x) = \sum_{r|(d,x)} r \mu(d/r)$. This is really not that difficult to get programmed in Mathematica right away. Please send me (the author) a personal email if you cannot figure out how to do this.