Exploring the Divisor Function

This note is part of the set Exploring the Divisor Function.

In this set, we aim to get a general form for this sum:

$\sum _{n=1}^{\infty}\frac{d\left(kn\right)}{n^2}$

Where $k$ 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 $f(n)$ is completely multiplicative, that is $f(ab)=f(a)f(b)$ , then

$f*f(n)=d(n)f(n)$

$\left[\sum _{n=1}^{ \infty}\frac{f\left(n\right)}{n^s}\right]^2=\sum _{n=1}^{\infty }\frac{f\left(n\right)d\left(n\right)}{n^s}$

Where $*$ is the Dirichlet Convolution

and $d(n)$ counts the number of divisors n.

I would post the solutions for the problems soon.

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Exploring the Divisor Function

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