Definition:Arithmetic Function

Definition

An arithmetic function or arithmetical function is a complex valued function defined on the strictly positive integers.

That is, a mapping:

$f: \Z_{>0} \to \C$

Also see

• Definition:Sequence, of which an arithmetic function is a particular instance

• Definition:Dirichlet Series
• Definition:Additive Arithmetic Function
• Definition:Multiplicative Arithmetic Function
• Definition:Dirichlet Convolution

• Results about arithmetic functions can be found here.

Examples

• The Möbius function: $\map \mu n$

• The Euler $\phi$ function: $\map \phi n$

• The von Mangoldt function: $\map \Lambda n$

Linguistic Note

In the context of an arithmetic function, the word arithmetic is pronounced with the stress on the first and third syllables: a-rith-me-tic, rather than on the second syllable: a-rith-me-tic.

This is because the word is being used in its adjectival form.

Sources

• 1976: Tom M. Apostol: Introduction to Analytic Number Theory ... (previous) ... (next): $2.1$: Introduction
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arithmetic function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic function