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:

$\sigma$ 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 $1$ .
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 $\mu$ .
$\sigma *\phi =E*E$ : The Dirichlet product of $\sigma$ and the Euler phi-function equals the Dirichlet product of the identity function with itself, which in turn is the (pointwise) product of the identity function and the divisor count function.