# Definition:Big-O Notation/Sequence

## Definition

### Definition 1

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \OO g$ is defined as:

$\map \OO g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le \size {\map f n} \le c \cdot \size {\map g n} }$

### Definition 2

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.

$a_n$ is big-$\OO$ of $b_n$ if and only if:

$\exists c \in \R_{\ge 0}: \exists n_0 \in \N: \forall n \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$

That is:

$\size {a_n} \le c \cdot \size {b_n}$

for all sufficiently large $n$.

This is denoted:

$a_n \in \map \OO {b_n}$

## Notation

The expression $\map f n \in \map \OO {\map g n}$ is read as:

$\map f n$ is big-O of $\map g n$

While it is correct and accurate to write:

$\map f n \in \map \OO {\map g n}$

it is a common abuse of notation to write:

$\map f n = \map \OO {\map g n}$

## Also defined as

Some authors require that $b_n$ be nonzero for $n$ sufficiently large.

Some authors require that the functions appearing in the $\OO$-estimate be positive or strictly positive.

## Also denoted as

The big-$\OO$ notation, along with little-$\oo$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.

In analytic number theory, sometimes Vinogradov's notations $f \ll g$ or $g \gg f$ are used to mean $f = \map \OO g$.

This can often be clearer for estimates leading to typographically complex error terms.

Some sources use an ordinary $O$:

$f = \map O g$