This note is part of the set Exploring the Divisor Function.
In this set, we aim to get a general form for this sum:
Where is a positive integer.
So, instead of giving out everything on a note, why not split it up into several problems so that everybody can try it out by themselves?
I will give a clue here, and then you can go ahead to solve the first problem of this set, slowly progressing to the last problem, where you will finally be able to find a general form of the sum. You may skip steps, because your approach might be better than mine. If you do have a better approach, do post it!
Here's the first clue:
If is completely multiplicative, that is , then
Where is the Dirichlet Convolution
and counts the number of divisors n.
I would post the solutions for the problems soon.