# Difference between revisions of "Mathematicial notation"

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+ | This is a list of '''symbols and conventions''' in mathematical notation. | ||

== Sets == | == Sets == | ||

− | <math> | + | <math>\mathbb{Z}</math>: the [[integer]]s (a [[unique factorization domain]]). |

<math>\mathbb{N}</math>: the [[natural number]]s. Unfortunately, this notation is ambiguous -- some authors use it for the [[positive integer]]s, some for the [[nonnegative integer]]s. | <math>\mathbb{N}</math>: the [[natural number]]s. Unfortunately, this notation is ambiguous -- some authors use it for the [[positive integer]]s, some for the [[nonnegative integer]]s. | ||

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When <math>M</math> is one of the sets from above, then <math>M^+</math> denotes the numbers <math>>0</math> (when defined), analogous for <math>M^-</math>. | When <math>M</math> is one of the sets from above, then <math>M^+</math> denotes the numbers <math>>0</math> (when defined), analogous for <math>M^-</math>. | ||

− | The meaning of <math>M^*</math> will depend on <math>M</math>: for most cases it denotes the invertible elements, but for <math> | + | The meaning of <math>M^*</math> will depend on <math>M</math>: for most cases it denotes the invertible elements, but for <math>\mathbb{Z}</math> it means the nonzero integers (note that these definitions coincide in most cases). |

A zero in the index, like in <math>M_0^+</math>, tells us that <math>0</math> is also included. | A zero in the index, like in <math>M_0^+</math>, tells us that <math>0</math> is also included. | ||

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<math>a</math> divides <math>b</math> (both integers) is written as <math>a|b</math>, or sometimes as <math>b \vdots a</math>. | <math>a</math> divides <math>b</math> (both integers) is written as <math>a|b</math>, or sometimes as <math>b \vdots a</math>. | ||

− | Then for <math>m,n \in \mathbb{Z}</math>, <math>\gcd(m,n)</math> or <math>(m,n)</math> is their '''greatest common divisor''', the greatest <math>d \in \mathbb{Z}</math> with <math> | + | Then for <math>m,n \in \mathbb{Z}</math>, <math>\gcd(m,n)</math> or <math>(m,n)</math> is their '''greatest common divisor''', the greatest <math>d \in \mathbb{Z}</math> with <math>d|m</math> and <math>d|n</math> (<math>\gcd(0,0)</math> is defined as <math>0</math>) and <math>\mathrm{lcm}(m,n)</math> or <math>\left[ m,n\right]</math> denotes their [[least common multiple]], the smallest non-negative integer <math>d</math> such that <math>m|d</math> and <math>n|d</math> |

. | . | ||

− | When <math> | + | When <math>\gcd(m,n)=1</math>, one often says that <math>m,n</math> are called "[[coprime]]". |

For <math>n \in \mathbb{Z}^*</math> to be '''squarefree''' means that there is no integer <math>k>1</math> with <math>k^2|n</math>. Equivalently, this means that no prime factor occurs more than once in the decomposition. | For <math>n \in \mathbb{Z}^*</math> to be '''squarefree''' means that there is no integer <math>k>1</math> with <math>k^2|n</math>. Equivalently, this means that no prime factor occurs more than once in the decomposition. | ||

− | '''Factorial''' of <math>n</math>: <math> | + | '''Factorial''' of <math>n</math>: <math>n! : = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1</math> |

− | '''Binomial Coefficients''': <math> | + | '''Binomial Coefficients''': <math>{n\choose k} = \frac{n!}{k! (n-k)!}</math> |

For two functions <math>f,g: \mathbb{N} \to \mathbb{C}</math> the '''Dirichlet convolution''' <math>f*g</math> is defined as <math>f*g(n) : = \sum_{d|n} f(d) g\left(\frac{n}{d}\right)</math>. | For two functions <math>f,g: \mathbb{N} \to \mathbb{C}</math> the '''Dirichlet convolution''' <math>f*g</math> is defined as <math>f*g(n) : = \sum_{d|n} f(d) g\left(\frac{n}{d}\right)</math>. | ||

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'''Euler's totient function''': <math>\varphi (n) = \phi (n) : = \left| \{ k \in \mathbb{N} \ : \ k \leq n , \gcd(k,n) \} \right| = \left| \mathbb{Z}_n^* \right|</math>. | '''Euler's totient function''': <math>\varphi (n) = \phi (n) : = \left| \{ k \in \mathbb{N} \ : \ k \leq n , \gcd(k,n) \} \right| = \left| \mathbb{Z}_n^* \right|</math>. | ||

− | '''Möbius' function''': <math>\mu(n): = \begin{cases} 0 & \textrm{ | + | '''Möbius' function''': <math>\mu(n): = \begin{cases} 0 & \textrm{ if } n\; \textrm{ is not squarefree} \\ (-1)^s & \textrm{ where } s \;\textrm{ is the number of prime factors of } n \;\textrm{ otherwise} \end{cases}</math>. |

'''Sum of powers of divisors''': <math>\sigma_k(n) : = \sum_{d|n} d^k</math>; often <math>\tau</math> is used for <math>\sigma_0</math>, the number of divisors, and simply <math>\sigma</math> for <math>\sigma_1</math>. | '''Sum of powers of divisors''': <math>\sigma_k(n) : = \sum_{d|n} d^k</math>; often <math>\tau</math> is used for <math>\sigma_0</math>, the number of divisors, and simply <math>\sigma</math> for <math>\sigma_1</math>. | ||

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With counting functions, some types of densities can be defined: | With counting functions, some types of densities can be defined: | ||

− | '''Lower asymptotic density''': <math> | + | '''Lower asymptotic density''': <math>_Ld(A) : =\liminf_{n \to \infty} \frac{a(n)}{n}</math> |

− | '''Upper asymptotic density''': <math> | + | '''Upper asymptotic density''': <math>_Ud(A) : =\limsup_{n \to \infty} \frac{a(n)}{n}</math> |

− | '''Asymptotic density''' (does not always exist): <math> | + | '''Asymptotic density''' (does not always exist): <math>d(A) : =\lim_{n \to \infty} \frac{a(n)}{n}</math> |

− | '''Shnirelman's density''': <math> | + | '''Shnirelman's density''': <math>\sigma(A) : =\inf_{n \to \infty} \frac{a(n)}{n}</math> |

− | '''Dirichlet's density'''(does not always exist): <math> | + | '''Dirichlet's density'''(does not always exist): <math>\delta(A) : = \lim_{s \to 1+0} \frac{\sum_{a \in A} a^{-s}}{\sum_{a \in \mathbb{N}} a^{-s}}</math> |

− | <math> | + | <math>{}_Ld(A)</math> and <math>_Ud(A)</math> are equal iff the asymptotic density <math>d(A)</math> exists and all three are equal then and equal to Dirichlet's density. |

Often, '''density''' is meant '''in relation to some other set''' <math>B</math> (often the primes). Then we need <math>A \subset B \subset \mathbb{N}</math> with counting functions <math> a,b </math> and simply change <math>n</math> into <math>b(n)</math> and <math>\mathbb{N}</math> into <math>B</math>: | Often, '''density''' is meant '''in relation to some other set''' <math>B</math> (often the primes). Then we need <math>A \subset B \subset \mathbb{N}</math> with counting functions <math> a,b </math> and simply change <math>n</math> into <math>b(n)</math> and <math>\mathbb{N}</math> into <math>B</math>: | ||

− | '''Lower asymptotic density''': <math> | + | '''Lower asymptotic density''': <math>_Ld_B(A) : =\liminf_{n \to \infty} \frac{a(n)}{b(n)} </math> |

− | '''Upper asymptotic density''': <math> | + | '''Upper asymptotic density''': <math>_Ud_B(A) : =\limsup_{n \to \infty} \frac{a(n)}{b(n)} </math> |

− | '''Asymptotic density''' (does not always exist): <math> | + | '''Asymptotic density''' (does not always exist): <math> d_B(A) : =\lim_{n \to \infty}{} \frac{a(n)}{b(n)} </math> |

− | '''Shnirelman's density''': <math> | + | '''Shnirelman's density''': <math>\sigma_B(A) : =\inf_{n \to \infty} \frac{a(n)}{b(n)} </math> |

− | '''Dirichlet's density'''(does not always exist): <math> | + | '''Dirichlet's density'''(does not always exist): <math>\delta_B(A) : = \lim_{s \to 1+0} \frac{\sum_{a \in A} a^{-s}}{\sum_{a \in B} a^{-s}} </math> |

− | Again the same relations as above hold. | + | Again, the same relations as above hold. |

## Latest revision as of 12:11, 17 June 2008

This is a list of **symbols and conventions** in mathematical notation.

## Sets

: the integers (a unique factorization domain).

: the natural numbers. Unfortunately, this notation is ambiguous -- some authors use it for the positive integers, some for the nonnegative integers.

: Also an ambiguous notation, use for the positive primes or the positive integers.

: the reals (a field).

: the complex numbers (an algebraically closed and complete field).

: the -adic numbers (a complete field); also and are used sometimes.

: the residues (a ring; a field for prime).

When is one of the sets from above, then denotes the numbers (when defined), analogous for . The meaning of will depend on : for most cases it denotes the invertible elements, but for it means the nonzero integers (note that these definitions coincide in most cases). A zero in the index, like in , tells us that is also included.

## Definitions

For a set , denotes the number of elements of .

divides (both integers) is written as , or sometimes as .
Then for , or is their **greatest common divisor**, the greatest with and ( is defined as ) and or denotes their least common multiple, the smallest non-negative integer such that and
.
When , one often says that are called "coprime".

For to be **squarefree** means that there is no integer with . Equivalently, this means that no prime factor occurs more than once in the decomposition.

**Factorial** of :

**Binomial Coefficients**:

For two functions the **Dirichlet convolution** is defined as .
A (weak) **multiplicative function** is one such that for all with .

Some special types of such functions:

**Euler's totient function**: .

**Möbius' function**: .

**Sum of powers of divisors**: ; often is used for , the number of divisors, and simply for .

For any it denotes the **number of representations of as sum of squares**.

Let be coprime integers. Then , the "**order of **" is the smallest with .

For and , the **-adic valuation ** can be defined as the multiplicity of in the factorisation of , and can be extended for by .
Additionally often is used.

For any function we define as the (upper) finite difference of . Then we set and then iteratively for all integers .

**Legendre symbol**: for and odd we define

Then the **Jacobi symbol** for and odd (prime factorization of ) is defined as:

**Hilbert symbol**: let and . Then
is the "Hilbert symbol of in respect to " (nontrivial means here that not all numbers are ).

When , then we can define a **counting function** .
One special case of a counting function is the one that belongs to the primes , which is often called .
With counting functions, some types of densities can be defined:

**Lower asymptotic density**:

**Upper asymptotic density**:

**Asymptotic density** (does not always exist):

**Shnirelman's density**:

**Dirichlet's density**(does not always exist):

and are equal iff the asymptotic density exists and all three are equal then and equal to Dirichlet's density.

Often, **density** is meant **in relation to some other set** (often the primes). Then we need with counting functions and simply change into and into :

**Lower asymptotic density**:

**Upper asymptotic density**:

**Asymptotic density** (does not always exist):

**Shnirelman's density**:

**Dirichlet's density**(does not always exist):

Again, the same relations as above hold.