Definition:Divisor Function
Definition
Let $\alpha \in \Z_{\ge 0}$ be a non-negative integer.
A divisor function is an arithmetic function of the form:
- $\ds \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$
where the summation is taken over all $m \le n$ such that $m$ divides $n$).
There exist the following special cases:
Divisor Count Function
Let $n$ be an integer such that $n \ge 1$.
The divisor count function is defined on $n$ as being the total number of positive integer divisors of $n$.
It is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\sigma_0$ (the Greek letter sigma).
That is:
- $\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Divisor Sum Function
Let $n$ be an integer such that $n \ge 1$.
The divisor sum function $\map {\sigma_1} n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.
That is:
- $\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Also known as
A divisor function is often seen referred to as a sigma ($\sigma$) function.
Also see
- Results about divisor functions can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sigma function or $\sigma$ function: 2.