# Symbols:Greek/Sigma

## Sigma

The $18$th letter of the Greek alphabet.

- Minuscules: $\sigma$ and $\varsigma$

- Majuscule: $\Sigma$

The $\LaTeX$ code for \(\sigma\) is `\sigma`

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The $\LaTeX$ code for \(\varsigma\) is `\varsigma`

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The $\LaTeX$ code for \(\Sigma\) is `\Sigma`

.

### Event Space

- $\Sigma$

Let $\EE$ be an experiment whose probability space is $\struct {\Omega, \Sigma, \Pr}$.

The **event space** of $\EE$ is usually denoted $\Sigma$ (Greek capital **sigma**), and is **the set of all outcomes of $\EE$ which are interesting**.

By definition, $\struct {\Omega, \Sigma}$ is a measurable space.

Hence the **event space** $\Sigma$ is a sigma-algebra on $\Omega$.

### Summation

Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.

The composite is called the **summation** of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

- $\ds \sum_{j \mathop = 1}^n a_j = \tuple {a_1 + a_2 + \cdots + a_n}$

The $\LaTeX$ code for \(\ds \sum_{j \mathop = 1}^n a_j\) is `\ds \sum_{j \mathop = 1}^n a_j`

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The $\LaTeX$ code for \(\ds \sum_{1 \mathop \le j \mathop \le n} a_j\) is `\ds \sum_{1 \mathop \le j \mathop \le n} a_j`

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The $\LaTeX$ code for \(\ds \sum_{\map \Phi j} a_j\) is `\ds \sum_{\map \Phi j} a_j`

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### Divisor Function

- $\map {\sigma_\alpha} n$

The **divisor function**:

- $\ds \map {\sigma_\alpha} n = \sum_{m \mathop \divides n} m^\alpha$

(meaning the summation is taken over all $m \le n$ such that $m$ divides $n$).

The $\LaTeX$ code for \(\map {\sigma_\alpha} n\) is `\map {\sigma_\alpha} n`

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### Divisor Counting Function

- $\map {\sigma_0} n$

Let $n$ be an integer such that $n \ge 1$.

The **divisor counting function** is defined on $n$ as being the total number of positive integer divisors of $n$.

It is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\sigma_0$ (the Greek letter **sigma**).

That is:

- $\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$

where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.

The $\LaTeX$ code for \(\map {\sigma_0} n\) is `\map {\sigma_0} n`

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### Divisor Sum Function

- $\map {\sigma_1} n$

Let $n$ be an integer such that $n \ge 1$.

The **divisor sum function** $\map {\sigma_1} n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.

That is:

- $\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d$

where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.

The $\LaTeX$ code for \(\map {\sigma_1} n\) is `\map {\sigma_1} n`

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### Surface Charge Density

- $\sigma$

Denotes the surface charge density of a given body:

- $\sigma = \dfrac q A$

where:

- $q$ is the body's electric charge;
- $A$ is the body's area.

### Area Density

- $\sigma$

Sometimes used, although $\rho_A$ (Greek letter **rho**) is more common, to denote the area density of a given two-dimensional body:

- $\sigma = \dfrac m A$

where: