Divisor 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 $n$ be a natural number. The divisor sum function of $n$, denoted $\sigma (n)$, is defined in the following equivalent ways:

1. $\sigma$ is the Dirichlet product of the identity function on the natural numbers and the all-one function: the function sending every natural number to $1$.
2. We have $\sigma (n)={\sum }_{d|n}d$.

$\sigma$ is a multiplicative function but not a completely multiplicative function.