Difference between revisions of "Mathematicial notation"

(Definitions)
 
(10 intermediate revisions by 6 users not shown)
Line 1: Line 1:
<math>\displaystyle \mathbb{Z}</math>: the integers (a unique factorisation domain).
+
This is a list of '''symbols and conventions''' in mathematical notation.
 +
== Sets ==
  
<math>\mathbb{N}</math>: the positive integers, meaning those $>0$.
 
  
<math>\mathbb{P}</math>: the positive primes.
+
<math>\mathbb{Z}</math>: the [[integer]]s (a [[unique factorization domain]]).
  
<math>\mathbb{Q}</math>: the rationals (a field).
+
<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{R}</math>: the reals (a field).
+
<math>\mathbb{P}</math>: Also an ambiguous notation, use for the positive [[prime]]s or the positive integers.
  
<math>\mathbb{C}</math>: the complex numbers (a algebraically closed and complete field).
+
<math>\mathbb{Q}</math>: the [[rational]]s (a [[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> is used sometimes.
+
<math>\mathbb{R}</math>: the [[real]]s (a field).
 +
 
 +
<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{Z}_n = \mathbb{Z} / n \mathbb{Z}</math>: the residues <math>\mod n</math> (a ring; a field for <math>n</math> prime).
 
<math>\mathbb{Z}_n = \mathbb{Z} / n \mathbb{Z}</math>: the residues <math>\mod n</math> (a ring; a field for <math>n</math> prime).
  
 
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>\displaystyle \mathbb{Z}</math> it means the nonzero integers (note that this definitions coincide in most cases).
+
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 26: Line 28:
 
For a set <math>M</math>, <math>|M|=\# M</math> denotes the number of elements of <math>M</math>.
 
For a set <math>M</math>, <math>|M|=\# M</math> denotes the number of elements of <math>M</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>.
+
<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 [b]greatest common divisor[/b], 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>
+
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>\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 $n \in \mathbb{Z}^*$ to be "[b]squarefree[/b]" means that there is no integer $k>1$ with $k^2|n$. 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>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>
  
[b]factorial[/b] of $n$: $n! : = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$
+
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>.
[b]binomial coefficients[/b]: $\binom{n}{k} = \frac{n!}{k! (n-k)!}$
+
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>.
  
For two functions $f,g: \mathbb{N} \to \mathbb{C}$ the [b]Dirichlet convolution[/b] $f*g$ is defined as $f*g(n) : = \sum_{d|n} f(d) g\left(\frac{n}{d}\right)$.
 
A (weak) [b]multiplicative function[/b] $f: \mathbb{N} \to \mathbb{C}$ is one such that $f(a\cdot b) = f(a) \cdot f(b)$ for all $a,b \in \mathbb{N}$ with $\gcd(a,b)=1$.
 
 
Some special types of such functions:
 
Some special types of such functions:
[b]Euler's totient function[/b]: $\varphi (n) = \phi (n) : = \left| \{ k \in \mathbb{N} \ : \ k \leq n , \gcd(k,n) \} \right| = \left| \mathbb{Z}_n^* \right|$.
 
[b]Möbius' function[/b]: $\mu(n): = \begin{cases} 0 \text{ iff } n \text{ is not squarefree} \\ (-1)^s \text{ where } s \text{ is the number of prime factors of } n \text{ otherwise} \end{cases}$.
 
[b]Sum of powers of divisors[/b]: $\sigma_k(n) : = \sum_{d|n} d^k$; often $\tau$ is used for $\sigma_0$, the number of divisors, and simply $\sigma$ for $\sigma_1$.
 
  
For any $k,n \in \mathbb{N}$ it denotes $r_k(n) : = \left| \{ (a_1,a_2,...,a_k) \in \mathbb{Z}^k | \sum a_i^2 = n \} \right|$ the [b]number of representations of $n$ as sum of $k$ squares[/b].
+
'''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>.
  
Let $a,n$ be coprime integers. Then $ord_n(a)$, the "[b]order of $a \mod n$[/b]" is the smallest $k \in \mathbb{N}$ with $a^k \equiv 1 \mod n$.
+
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 $n \in \mathbb{Z}^*$ and $p \in \mathbb{P}$, the [b]$p$-adic valuation $v_p(n)$[/b] can be defined as the multiplicity of $p$ in the factorisation of $n$, and can be extended for $\frac{m}{n} \in \mathbb{Q}^* , \ m,n \in \mathbb{Z}^*$ by $v_p\left( \frac{m}{n} \right) = v_p(m)-v_p(n)$.
+
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>.
Additionally often $v_p(0) = \infty$ is used.
+
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>.
  
For any function $f$ we define $\Delta (f)(x) : = f(x+1)-f(x)$ as the (upper) finite difference of $f$.
 
Then we set $\Delta^0(f)(x) : = f(x)$ and then iteratively $\Delta^n (f) (x) : = \Delta(\Delta^{n-1} (f)) (x)$ for all integers $n \geq 1$.
 
  
 +
'''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>
  
[b]Legendre symbol:[/b] for $a \in \mathbb{Z}$ and odd $p \in \mathbb{P}$ we define $\left( \frac{a}{p} \right) : = \begin{cases} 1 & \text{ when } x^2 \equiv a \mod p \text{ has a solution } x \in \mathbb{Z}_p^* \\ 0 & \text{ iff } p|a \\ -1 & \text{ when } x^2 \equiv a \mod p \text{ has no solution } x \in \mathbb{Z}_p \end{cases}$
+
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 [b]Jacobi symbol[/b] for $a \in \mathbb{Z}$ and odd $n= \prod p_i^{v_i}$ (prime factorisation of $n$) is defined as: $\left( \frac{a}{n} \right) = \prod \left( \frac{a}{p_i} \right)^{v_i}$
 
  
[b]Hilbert symbol[/b]: let $v \in \mathbb{P} \cup \{ 0 , \infty \}$ and $a,b \in \mathbb{Q}_v^*$. Then
+
'''Hilbert symbol''': let <math>v \in \mathbb{P} \cup \{ 0 , \infty \}</math> and <math>a,b \in \mathbb{Q}_v^*</math>. Then
\[ \left( a , b \right)_v : = \begin{cases} 1 & \text{ iff } x^2=ay^2+bz^2 \text{ has a nontrivial solution } (x,y,z) \in \mathbb{Q}_v^3 \\ -1 & \text{ otherwise} \end{cases}  \]
+
<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 $a,b$ in respect to $v$" (nontrivial means here that not all numbers are $0$).
+
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 $A \subset \mathbb{N}$, then we can define a [b]counting function[/b] $a(n) : = | \{ a \in A | a \leq n \}$.
+
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 $\mathbb{P}$, which is often called $\pi$.
+
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:
  
[b]Lower asymptotic density[/b]: $_Ld(A) : =\liminf_{n \to \infty} \frac{a(n)}{n}$
+
'''Lower asymptotic density''': <math>_Ld(A) : =\liminf_{n \to \infty} \frac{a(n)}{n}</math>
[b]Upper asymptotic density[/b]: $_Ud(A) : =\limsup_{n \to \infty} \frac{a(n)}{n}$
+
 
[b]Asymptotic density[/b] (does not always exist): $d(A) : =\lim_{n \to \infty} \frac{a(n)}{n}$
+
'''Upper asymptotic density''': <math>_Ud(A) : =\limsup_{n \to \infty} \frac{a(n)}{n}</math>
[b]Shnirelman's density[/b]: $\sigma(A) : =\inf_{n \to \infty} \frac{a(n)}{n}$
+
 
[b]Dirichlet's density[/b](does not always exist): $\delta(A) : = \lim_{s \to 1+0} \frac{\sum_{a \in A} a^{-s}}{\sum_{a \in \mathbb{N}} a^{-s}}$
+
'''Asymptotic density''' (does not always exist): <math>d(A) : =\lim_{n \to \infty} \frac{a(n)}{n}</math>
$_Ld(A)$ and $_Ud(A)$ are equal iff the asymptotic density $d(A)$ exists and all three are equal then and equal to Dirichlet's density.
+
 
 +
'''Shnirelman's density''': <math>\sigma(A) : =\inf_{n \to \infty} \frac{a(n)}{n}</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>{}_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>:
 +
 
 +
'''Lower asymptotic density''': <math>_Ld_B(A) : =\liminf_{n \to \infty} \frac{a(n)}{b(n)} </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>  d_B(A) : =\lim_{n \to \infty}{} \frac{a(n)}{b(n)} </math>
  
 +
'''Shnirelman's density''': <math>\sigma_B(A) : =\inf_{n \to \infty} \frac{a(n)}{b(n)} </math>
  
Often, [b]density[/b] is meant [b]in relation to some other set[/b] $B$ (often the primes). Then we need $A \subset B \subset \mathbb{N}$ with counting functions $a,b$ and simply change $n$ into $b(n)$ and $\mathbb{N}$ into $B$:
+
'''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>
  
[b]Lower asymptotic density[/b]: $_Ld_B(A) : =\liminf_{n \to \infty} \frac{a(n)}{b(n)}$
+
Again, the same relations as above hold.
[b]Upper asymptotic density[/b]: $_Ud_B(A) : =\limsup_{n \to \infty} \frac{a(n)}{b(n)}$
 
[b]Asymptotic density[/b] (does not always exist): $d_B(A) : =\lim_{n \to \infty} \frac{a(n)}{b(n)}$
 
[b]Shnirelman's density[/b]: $\sigma_B(A) : =\inf_{n \to \infty} \frac{a(n)}{b(n)}$
 
[b]Dirichlet's density[/b](does not always exist): $\delta_B(A) : = \lim_{s \to 1+0} \frac{\sum_{a \in A} a^{-s}}{\sum_{a \in B} a^{-s}}$
 
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

$\mathbb{Z}$: the integers (a unique factorization domain).

$\mathbb{N}$: the natural numbers. Unfortunately, this notation is ambiguous -- some authors use it for the positive integers, some for the nonnegative integers.

$\mathbb{P}$: Also an ambiguous notation, use for the positive primes or the positive integers.

$\mathbb{Q}$: the rationals (a field).

$\mathbb{R}$: the reals (a field).

$\mathbb{C}$: the complex numbers (an algebraically closed and complete field).

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

$\mathbb{Z}_n = \mathbb{Z} / n \mathbb{Z}$: the residues $\mod n$ (a ring; a field for $n$ prime).

When $M$ is one of the sets from above, then $M^+$ denotes the numbers $>0$ (when defined), analogous for $M^-$. The meaning of $M^*$ will depend on $M$: for most cases it denotes the invertible elements, but for $\mathbb{Z}$ it means the nonzero integers (note that these definitions coincide in most cases). A zero in the index, like in $M_0^+$, tells us that $0$ is also included.

Definitions

For a set $M$, $|M|=\# M$ denotes the number of elements of $M$.

$a$ divides $b$ (both integers) is written as $a|b$, or sometimes as $b \vdots a$. Then for $m,n \in \mathbb{Z}$, $\gcd(m,n)$ or $(m,n)$ is their greatest common divisor, the greatest $d \in \mathbb{Z}$ with $d|m$ and $d|n$ ($\gcd(0,0)$ is defined as $0$) and $\mathrm{lcm}(m,n)$ or $\left[ m,n\right]$ denotes their least common multiple, the smallest non-negative integer $d$ such that $m|d$ and $n|d$ . When $\gcd(m,n)=1$, one often says that $m,n$ are called "coprime".

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


Factorial of $n$: $n! : = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$

Binomial Coefficients: ${n\choose k} = \frac{n!}{k! (n-k)!}$

For two functions $f,g: \mathbb{N} \to \mathbb{C}$ the Dirichlet convolution $f*g$ is defined as $f*g(n) : = \sum_{d|n} f(d) g\left(\frac{n}{d}\right)$. A (weak) multiplicative function $f: \mathbb{N} \to \mathbb{C}$ is one such that $f(a\cdot b) = f(a) \cdot f(b)$ for all $a,b \in \mathbb{N}$ with $\gcd(a,b)=1$.

Some special types of such functions:

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

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

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

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

Let $a,n$ be coprime integers. Then $ord_n(a)$, the "order of $a \mod n$" is the smallest $k \in \mathbb{N}$ with $a^k \equiv 1 \mod n$.

For $n \in \mathbb{Z}^*$ and $p \in \mathbb{P}$, the $p$-adic valuation $v_p(n)$ can be defined as the multiplicity of $p$ in the factorisation of $n$, and can be extended for $\frac{m}{n} \in \mathbb{Q}^* , \ m,n \in \mathbb{Z}^*$ by $v_p\left( \frac{m}{n} \right) = v_p(m)-v_p(n)$. Additionally often $v_p(0) = \infty$ is used.

For any function $f$ we define $\Delta (f)(x) : = f(x+1)-f(x)$ as the (upper) finite difference of $f$. Then we set $\Delta^0(f)(x) : = f(x)$ and then iteratively $\Delta^n (f) (x) : = \Delta(\Delta^{n-1} (f)) (x)$ for all integers $n \geq 1$.


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

Then the Jacobi symbol for $a \in \mathbb{Z}$ and odd $n= \prod p_i^{\nu_i}$ (prime factorization of $n$) is defined as: $\left( \frac{a}{n} \right) = \prod \left( \frac{a}{p_i} \right)^{\nu_i}$

Hilbert symbol: let $v \in \mathbb{P} \cup \{ 0 , \infty \}$ and $a,b \in \mathbb{Q}_v^*$. Then $\left( a , b \right)_v : = \begin{cases} 1 & \textrm{ iff } x^2=ay^2+bz^2 \textrm{ has a nontrivial solution } (x,y,z) \in \mathbb{Q}_v^3 \\ -1 & \textrm{ otherwise} \end{cases}$ is the "Hilbert symbol of $a,b$ in respect to $v$" (nontrivial means here that not all numbers are $0$).


When $A \subset \mathbb{N}$, then we can define a counting function $a(n) : = | \{ a \in A | a \leq n \}$. One special case of a counting function is the one that belongs to the primes $\mathbb{P}$, which is often called $\pi$. With counting functions, some types of densities can be defined:

Lower asymptotic density: $_Ld(A) : =\liminf_{n \to \infty} \frac{a(n)}{n}$

Upper asymptotic density: $_Ud(A) : =\limsup_{n \to \infty} \frac{a(n)}{n}$

Asymptotic density (does not always exist): $d(A) : =\lim_{n \to \infty} \frac{a(n)}{n}$

Shnirelman's density: $\sigma(A) : =\inf_{n \to \infty} \frac{a(n)}{n}$

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

${}_Ld(A)$ and $_Ud(A)$ are equal iff the asymptotic density $d(A)$ exists and all three are equal then and equal to Dirichlet's density.


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

Lower asymptotic density: $_Ld_B(A) : =\liminf_{n \to \infty} \frac{a(n)}{b(n)}$

Upper asymptotic density: $_Ud_B(A) : =\limsup_{n \to \infty} \frac{a(n)}{b(n)}$

Asymptotic density (does not always exist): $d_B(A) : =\lim_{n \to \infty}{} \frac{a(n)}{b(n)}$

Shnirelman's density: $\sigma_B(A) : =\inf_{n \to \infty} \frac{a(n)}{b(n)}$

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

Again, the same relations as above hold.

Invalid username
Login to AoPS