Convolution

For two functions $f, g : \mathbb{N} \rightarrow \mathbb{C}$ , the Dirichlet convolution (or simply convolution) $\displaystyle f * g$ of $\displaystyle f$ and $\displaystyle g$ is defined as

$\sum_{d\mid n} f(d)g\left( \frac{n}{d} \right)$ .

We usually only consider positive divisors of $\displaystyle n$ . We are often interested in convolutions of weak multiplicative functions; the set of weak multiplicative functions is closed under convolution. In general, convolution is commutative and associative; it also has an identity, the function $\displaystyle f(n)$ defined to be 1 if $\displaystyle n=1$ , and 0 otherwise. However, not all functions have inverses (e.g., the function $\displaystyle f(n) : n \mapsto 0$ has no inverse, as $\displaystyle f*g = f$ , for all functions $g: \mathbb{N} \rightarrow \mathbb{C}$ ), although many do.

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Convolution