Divisor power sum function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Let $k$ be a real number (typically an integer). The divisor power sum function $σ_{k}$ is defined as the following arithmetic function from the natural numbers to the real numbers:

$σ_{k} (n) = \sum_{d | n} d^{k}$ .

The sum is over all the positive divisors of $n$ .

Definition in terms of Dirichlet product

The divisor power sum function is defined as:

$σ_{k} : = E_{k} * U$ .

Here $E_{k}$ is the $k^{t h}$ power function, and $U$ is the all ones function.

Particular cases

The $k = 0$ case

The case $k = 0$ gives the divisor count function, i.e., the function that counts the number of positive divisors of $n$ .

The $k = 1$ case

The case $k = 1$ gives the divisor sum function, i.e., the sum of all the positive divisors.