Arithmetic Functions

An arithmetic function is a function ${\displaystyle {\mathbb{Z}}^{+}\to S}$ where ${\displaystyle S\subseteq ℝ\subseteq ℂ}$.  They try to capture the essence of various properties of positive whole numbers.

Examples
The constant ${\displaystyle 1}$ function maps every  n  to the number 1.

The identity function ${\displaystyle I}$ maps every  n  to itself.

Möbius function ${\displaystyle \mu }$

The unit function ${\displaystyle {\delta }_1}$ (or ${\displaystyle \epsilon }$ function)

Euler's totient function ${\displaystyle \phi }$

Number of divisors ${\displaystyle \tau }$

Sum of divisors ${\displaystyle \tau }$