# Multiple Dirichlet convolutions

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).

Convoluting $m$ copies of $1$ (the constant function) counts the number of $m$-length divisor paths in the divisor poset of $n$. Or equivalently the number of ways of writing $n = a_1 \cdots a_m$ where $a_i \mid n$.
For $n = 0$ the only path is $n$ itself, so there is one path.
For $n = 1, a_n = 3$ so $1 * 1 * 1 = \sum_{a'b' = n}1(b')\sum_{ab = a'} 1(a)1(b) = \sum_{abc = n} 1$.