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).
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Definition

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

$\sigma _{k}(n)=\sum _{d|n}d^{k}$.

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

Definition in terms of Dirichlet product

The divisor power sum function is defined as:

$\sigma _{k}:=E_{k}*U$.

Here $E_{k}$ is the $k^{th}$ 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.

Dirichlet series

Further information: Formula for Dirichlet series of divisor power sum function

The Dirichlet series for $\sigma _{k}$ is given by:

$\sum _{n\in \mathbb {N} }{\frac {\sigma _{k}(n)}{n^{s}}}$.

This is related to the Riemann zeta-function by the following identity, that holds both formally and for the corresponding meromorphic functions:

$\sum _{n\in \mathbb {N} }{\frac {\sigma _{k}(n)}{n^{s}}}=\zeta (s)\zeta (s+k)$.