Dirichlet convolution

Definition

Let $f$ and $g$ be arithmetic functions. Their Dirichlet convolution is the arithmetic function $f * g$ given by $$(f * g)(n) = \sum_{d \mid n} f(d) \, g\!\left(\frac{n}{d}\right)$$ for each positive integer $n$, where $d$ runs over the positive divisors of $n$.

Remarks

The convolution is commutative and associative, and it distributes over addition. The identity element is $\varepsilon$, with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$. An arithmetic function $f$ has an inverse for $*$ if and only if $f(1) \ne 0$. For example, with the constant function $\mathbf{1}$, $(\mathbf{1} * \mathbf{1})(n)$ is the number of divisors of $n$.

Explanations

L1 human

The Dirichlet convolution combines two functions by a sum over the factorizations $n = d \cdot e$. It is the natural product for functions defined by sums over divisors.