Arithmetic function

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 $\varphi$, the divisor function $\sigma_k$ and the Möbius function $\mu$. Some texts allow values in another commutative ring, or set $f(0) = 0$ and use all natural numbers as the domain. The Dirichlet convolution 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.