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).
View a complete list of arithmetic functions

Definition

Let $n$ be a natural number. The divisor sum function of $n$, denoted $\sigma (n)$ or ${\sigma }_1(n)$, 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:

$\sigma (n)=\prod _{i=1}^r(\frac{p_i^{k_i+1}-1}{p_i-1})$.

Equivalently, the ratio is given by:

$\frac{\sigma (n)}{n}=\prod _{i=1}^r(\frac{1-(1/p_i^{k_i+1})}{1-1/p_i})$.

Behavior

Lower bound

For any $n>1$, $\sigma (n)\ge n+1$. 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:

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

and:

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

Upper bound

Gronwall's theorem asserts that:

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

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 $n\ge 5041$, the ratio is always strictly less than $e^{\gamma }$.

Logarithmic ratio

We have:

$\underset{n\to \mathrm{\infty }}{lim}\frac{\log (\sigma (n))}{\log n}=1$.

Moreover, this approach is from the positive side, since $\sigma (n)\ge n+1$ for all $n>1$.

Summatory function and average value

Summatory function

The summatory function of this function is termed the divisor sum summatory function, and is defined as:

$x\mapsto {\sum }_{n\le x}\sigma (n)$.

It is equivalent to the following:

${\sum }_{d\le x}d[\frac{x}{d}]$.

In other words, it is the sum, over all numbers less than or equal to $x$, of the largest multiple of that number less than or equal to $x$. Note that this summatory function is bounded from above by $x^2$ and from below by $x^2/2$.

Average value

Given any positive real $x$, consider the ratio:

$\frac{{\sum }_{n\le x}\sigma (n)}{{\sum }_{n\le x}n}$.

This ratio is bounded from below by $1$ and from above by $2$.

Relation with other arithmetic functions

Generalizations

• Divisor power sum function ${\sigma }_k$ is the sum of the $k^{th}$ powers of all the positive divisors. The divisor sum function is the divisor power sum function ${\sigma }_1$. Also related is the divisor count function ${\sigma }_0$, which is also denoted $d$ or $\tau$.

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

• Prime number: A natural number $n$ such that $\sigma (n)=n+1$.
• Perfect number: A natural number $n$ such that $\sigma (n)=2n$.
• Quasiperfect number: A natural number $n$ such that $\sigma (n)=2n+1$.
• 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

Multiplicativity

This arithmetic function is a multiplicative function: the product of this function for two natural numbers that are relatively prime is the value of the function at the product.
View a complete list of multiplicative functions

The divisor sum function is multiplicative: in other words, if $m$ and $n$ are relatively prime positive integers, then:

$\sigma (mn)=\sigma (m)\sigma (n)$.

This can be proved in a number of ways. Apart from a direct proof, it also follows from the fact that the divisor sum function is a Dirichlet product of two multiplicative functions. For full proof, refer: Divisor sum function is multiplicative

Complete multiplicativity

NO: This arithmetic function is not a completely multiplicative function: in other words, the product of the values of the function at two natural numbers need not equal the value at the product.

The divisor sum function is not a completely multiplicative function. In other words, there exist natural numbers $m,n$ such that $\sigma (mn)\ne \sigma (m)\sigma (n)$. In fact, if $m$ and $n$ are not relatively prime, $\sigma (mn)$ is not the product of $\sigma (m)$ and $\sigma (n)$.

Preservation of divisibility

NO: This arithmetic function is not a divisibility-preserving function: it does not preserve divisibility.

If $m$ divides $n$, it is not necessary that $\sigma (m)$ divides $\sigma (n)$.

External links

• Divisor function on Mathworld