Dirichlet convolution

Builds on Arithmetic function Divisors of a natural number

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 $n$ .

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Tag	`0KH1N66QWW`
URI	`m2://0KH1N66QWW`
Aliases	`dirichlet-product`
Kind	definition
Writing	W1 stated
MSC	11A25
Defines	Dirichlet convolution
Uses	Arithmetic function, Divisors of a natural number

Dirichlet convolution

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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