I have been playing around with Dirichlet convolutions. As a reminder, take two arithmetic functions $f,g$, then their Dirichlet convolution is defined as the arithmetic function with:
$(f\star g)(n) = \sum_{d\mid n}f(d)g\left(\frac{n}{d}\right)$.
In particular I was wondering if anyone knows of any nice identities for multiple convolutions?
For example, take the arithmetic function $N_{\alpha}(n) = n^{\alpha}$ (for $\alpha\in\mathbb{Z}$).
Then, given $a,b\in\mathbb{Z}$ we have:
$N_a \star N_b = N_a \sigma_{b-a}$
and for any $n$-tuple $(a_1,a_2,...,a_n)\in\mathbb{Z}^n$ (for $n\geq 3$):
$(N_{a_1} \star N_{a_2} \star N_{a_3} ... \star N_{a_n}) = N_{a_1}(\sigma_{a_2-a_1} \star N_{a_3-a_1} \star N_{a_4-a_1} \star ... \star N_{a_n - a_1})$
(giving a recursion to handle multiple convolutions of $N$'s).