Using Dirichlet convolution where f = μ ∗ μ (Mobius) to find f(24)?

Question

I am confused about the Dirichlet convolution and how it is used. Does it take two entirely different arithmetic functions?

And knowing that f = μ ∗ μ (the Mobius function), why does the question I am solving suggest the Dirichlet convolution formula should be used to solve this, to find f(24)?

Answer 1

The Dircihlet convolution is defined for $g,h $ (arbitrary) arithmetic functions as $$(g \ast h) (n) = \sum_{d\mid n} g(d) h(n/d).$$ It can be thought of as some kind of product of arithmetic functions.

In this sense, yes, it can take entirely different functions $g$ and $h$ and create a new function $g\ast h$; just like you can take two polynomials and form their product. Of course $g=h$ is also possible.

Now, you want to consider $\mu \ast \mu$, and to calculate $(\mu \ast \mu)(24)$ you have
$$(\mu \ast \mu) (24) = \sum_{d\mid 24} \mu(d) \mu(24/d)$$ and then evaluate the right-hand side explicitly. (There may be more clever ways to evaluate this, but in fact it seems you are supposed to do this by this formula.)