Definition:Multiplicative Arithmetic Function
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This page is about Multiplicative Arithmetic Function. For other uses, see Multiplicative Function.
Definition
Let $f : \N \to \C$ be an arithmetic function.
Then $f$ is multiplicative if and only if:
- $m \perp n \implies \map f {m n} = \map f m \map f n$
where $m \perp n$ denotes that $m$ is coprime to $n$.
That is, a multiplicative arithmetic function is one where the value of a product of two coprime numbers equals the product of the value of each one individually.
Also see
- Definition:Additive Arithmetic Function
- Definition:Completely Additive Function
- Definition:Completely Multiplicative Function
- Results about multiplicative functions can be found here.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $29$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiplicative function: 2.