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:

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

Equivalently, the ratio is given by:

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

Behavior

Lower bound

For any $n>1$, $\sigma (n)\geq n+1$. Equality is achieved if and only if $n$ is prime. Further, $\sigma (n)$ is the lowest, in relative terms, for primes. In particular, excluding the case $n=1$, the fraction $\sigma (n)/n$ achieves a strict minimum-so-far at every prime, and nowhere else.

Thus, we have:

$\lim _{n\to \infty }\sigma (n)\to \infty$,

and:

$\lim \inf _{n\to \infty }{\frac {\sigma (n)}{n}}=1$,

Upper bound

Gronwall's theorem asserts that:

$\lim \sup _{n\to \infty }{\frac {\sigma (n)}{n\log \log n}}=e^{\gamma }$

where all the logarithms are natural logarithms and $\gamma$ is the Euler-Mascheroni constant.

A closely related result is Robin's theorem, which states that the Riemann hypothesis is equivalent to the statement that for $n\geq 5041$, the ratio is always strictly less than $e^{\gamma }$.

Summatory function

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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 ones 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.

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 $\sigma (n)<2n$.
• Superabundant number: A natural number $n$ that is a strict maximum-so-far for $\sigma (n)/n$.

Properties

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