# Difference between revisions of "Mathematicial notation"

(proofreading) |
(→Definitions) |
||

(7 intermediate revisions by 4 users not shown) | |||

Line 1: | Line 1: | ||

+ | 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 positive | + | <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{P}</math>: the positive | + | <math>\mathbb{P}</math>: Also an ambiguous notation, use for the positive [[prime]]s or the positive integers. |

− | <math>\mathbb{Q}</math>: the | + | <math>\mathbb{Q}</math>: the [[rational]]s (a [[field]]). |

− | <math>\mathbb{R}</math>: the | + | <math>\mathbb{R}</math>: the [[real]]s (a field). |

− | <math>\mathbb{C}</math>: the complex | + | <math>\mathbb{C}</math>: the [[complex number]]s (an [[algebraically closed]] and [[complete]] field). |

<math>\mathbb{Q}_p</math>: the <math>p</math>-adic numbers (a complete field); also <math>\mathbb{Q}_0 : =\mathbb{Q}</math> and <math>\mathbb{Q}_\infty : = \mathbb{R}</math> are used sometimes. | <math>\mathbb{Q}_p</math>: the <math>p</math>-adic numbers (a complete field); also <math>\mathbb{Q}_0 : =\mathbb{Q}</math> and <math>\mathbb{Q}_\infty : = \mathbb{R}</math> are used sometimes. | ||

Line 19: | Line 20: | ||

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. | ||

− | |||

− | |||

== Definitions == | == Definitions == | ||

Line 34: | Line 33: | ||

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

− | For | + | 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>n! : = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1</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>. | |

− | + | A (weak) '''multiplicative function''' <math>f: \mathbb{N} \to \mathbb{C}</math> is one such that <math>f(a\cdot b) = f(a) \cdot f(b)</math> for all <math>a,b \in \mathbb{N}</math> with <math>\gcd(a,b)=1</math>. | |

− | |||

− | |||

Some special types of such functions: | Some special types of such functions: | ||

− | |||

− | |||

− | |||

− | For any | + | '''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{ 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>. | ||

+ | |||

+ | For any <math>k,n \in \mathbb{N}</math> it denotes <math>r_k(n) : = \left| \{ (a_1,a_2,...,a_k) \in \mathbb{Z}^k | \sum a_i^2 = n \} \right|</math> the '''number of representations of <math>n</math> as sum of <math>k</math> squares'''. | ||

+ | |||

+ | Let <math>a,n</math> be coprime integers. Then <math>ord_n(a)</math>, the "'''order of <math>a \mod n</math>'''" is the smallest <math>k \in \mathbb{N}</math> with <math>a^k \equiv 1 \mod n</math>. | ||

− | + | For <math>n \in \mathbb{Z}^*</math> and <math>p \in \mathbb{P}</math>, the '''<math>p</math>-adic valuation <math>v_p(n)</math>''' can be defined as the multiplicity of <math>p</math> in the factorisation of <math>n</math>, and can be extended for <math>\frac{m}{n} \in \mathbb{Q}^* , \ m,n \in \mathbb{Z}^*</math> by <math>v_p\left( \frac{m}{n} \right) = v_p(m)-v_p(n)</math>. | |

+ | Additionally often <math>v_p(0) = \infty</math> is used. | ||

− | For | + | For any function <math>f</math> we define <math>\Delta (f)(x) : = f(x+1)-f(x)</math> as the (upper) finite difference of <math>f</math>. |

− | + | Then we set <math>\Delta^0(f)(x) : = f(x)</math> and then iteratively <math>\Delta^n (f) (x) : = \Delta(\Delta^{n-1} (f)) (x)</math> for all integers <math>n \geq 1</math>. | |

− | |||

− | |||

+ | '''Legendre symbol''': for <math>a \in \mathbb{Z}</math> and [[odd integer | odd]] <math>p \in \mathbb{P}</math> we define <math>\left( \frac{a}{p} \right) : = \begin{cases} 1 & \textrm{ when } x^2 \equiv a \mod p \textrm{ has a solution } x \in \mathbb{Z}_p^* \\ 0 & \textrm{ iff } p|a \\ -1 & \textrm{ when } x^2 \equiv a \mod p \textrm{ has no solution } x \in \mathbb{Z}_p \end{cases}</math> | ||

− | + | Then the '''Jacobi symbol''' for <math>a \in \mathbb{Z}</math> and odd <math>n= \prod p_i^{\nu_i}</math> (prime factorization of <math>n</math>) is defined as: <math>\left( \frac{a}{n} \right) = \prod \left( \frac{a}{p_i} \right)^{\nu_i}</math> | |

− | Then the | ||

− | + | '''Hilbert symbol''': let <math>v \in \mathbb{P} \cup \{ 0 , \infty \}</math> and <math>a,b \in \mathbb{Q}_v^*</math>. Then | |

− | + | <math> \left( a , b \right)_v : = \begin{cases} 1 & \textrm{ iff } x^2=ay^2+bz^2 \textrm{ has a nontrivial solution } (x,y,z) \in \mathbb{Q}_v^3 \\ -1 & \textrm{ otherwise} \end{cases} </math> | |

− | is the "Hilbert symbol of | + | is the "Hilbert symbol of <math>a,b</math> in respect to <math>v</math>" (nontrivial means here that not all numbers are <math>0</math>). |

− | When | + | When <math>A \subset \mathbb{N}</math>, then we can define a '''counting function''' <math>a(n) : = | \{ a \in A | a \leq n \}</math>. |

− | One special case of a counting function is the one that belongs to the primes | + | One special case of a counting function is the one that belongs to the primes <math>\mathbb{P}</math>, which is often called <math>\pi</math>. |

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''' | + | 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.