---
tag: 0H2EPRZC0D
kind: definition
name: Arithmetic function
aliases: [arithmetical-function, number-theoretic-function]
msc: [11A25]
defines: 09115NQZ07
sources:
  - url: https://en.wikipedia.org/wiki/Arithmetic_function
    locator: lead section
---

## Definition

An arithmetic function is a function $f \colon \mathbb{Z}_{>0} \to \mathbb{C}$ from the positive integers to the complex numbers.

## Remarks

Examples are [[Euler's totient function|0VV04M41AN]] $\varphi$, the [[divisor function|0YEW47VJ0J]] $\sigma_k$ and the [[Möbius function|0C8QENHYZW]] $\mu$.
Some texts allow values in another commutative ring, or set $f(0) = 0$ and use all natural numbers as the domain.
The [[Dirichlet convolution|0KH1N66QWW]] makes the arithmetic functions into a commutative ring.

## Explanations

### L1 human

An arithmetic function is a sequence $f(1), f(2), f(3), \dots$ of numbers, seen as a function of $n$.
Number theory studies such functions when they describe arithmetic properties of $n$, such as its number of divisors.