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

Formula in terms of prime factorization

Suppose we have:

$n=p_{1}^{k_{1}}p_{2}^{k_{2}}\dots p_{r}^{k_{r}}$,

where the $p_{i}$ are distinct prime divisors of $n$. Then:

$\sigma (n)=\prod _{i=1}^{r}\left({\frac {p_{i}^{k_{i}+1}-1}{p_{i}-1}}\right)$.

Relation with other arithmetic functions

Relations expressed in terms of Dirichlet products

• $\sigma =E*U$: $\sigma$ is the Dirichlet product of the identity function and the all-one function.
• $\sigma *\mu =E$: The Dirichlet product of $\sigma$ and the Mobius function is the identity function. Note that this is the [[Mobius inversion formula applied to the previous statement; equivalently, it is obtained by multiplying both sides of the previous equation by $\mu$.

Relation with properties of numbers

• Perfect number: A natural number $n$ such that $\sigma (n)=2n$.
• Abundant number: A natural number $n$ such that $\sigma (n)>2n$.
• Deficient number: A natural number $n$ such that $\sigma (n)<2n$.

Properties

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