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| − | This is a list of '''symbols and conventions''' in mathematical notation.
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| − | == Sets ==
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| − | <math>\mathbb{Z}</math>: the [[integer]]s (a [[unique factorization domain]]).
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| − | <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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| − | <math>\mathbb{P}</math>: Also an ambiguous notation, use for the positive [[prime]]s or the positive integers.
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| − | <math>\mathbb{Q}</math>: the [[rational]]s (a [[field]]).
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| − | <math>\mathbb{R}</math>: the [[real]]s (a field).
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| − | <math>\mathbb{C}</math>: the [[complex number]]s (an [[algebraically closed]] and [[complete]] field).
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| − | <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.
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| − | <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).
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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>.
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| − | 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).
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| − | A zero in the index, like in <math>M_0^+</math>, tells us that <math>0</math> is also included.
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| − | == Definitions ==
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| − | For a set <math>M</math>, <math>|M|=\# M</math> denotes the number of elements of <math>M</math>.
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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>.
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| − | 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>
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| − | .
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| − | When <math>\gcd(m,n)=1</math>, one often says that <math>m,n</math> are called "[[coprime]]".
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| − | 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.
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| − | '''Factorial''' of <math>n</math>: <math>n! : = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1</math>
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| − | '''Binomial Coefficients''': <math>{n\choose k} = \frac{n!}{k! (n-k)!}</math>
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| − | 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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| − | 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>.
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| − | Some special types of such functions:
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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>.
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| − | '''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>.
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| − | '''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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| − | 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'''.
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| − | 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>.
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| − | 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>.
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| − | Additionally often <math>v_p(0) = \infty</math> is used.
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| − | 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>.
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| − | 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>.
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| − | '''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>
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| − | 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>
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| − | '''Hilbert symbol''': let <math>v \in \mathbb{P} \cup \{ 0 , \infty \}</math> and <math>a,b \in \mathbb{Q}_v^*</math>. Then
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| − | <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>
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| − | 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>).
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| − | When <math>A \subset \mathbb{N}</math>, then we can define a '''counting function''' <math>a(n) : = | \{ a \in A | a \leq n \}</math>.
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| − | 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>.
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| − | With counting functions, some types of densities can be defined:
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| − | '''Lower asymptotic density''': <math>_Ld(A) : =\liminf_{n \to \infty} \frac{a(n)}{n}</math>
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| − | '''Upper asymptotic density''': <math>_Ud(A) : =\limsup_{n \to \infty} \frac{a(n)}{n}</math>
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| − | '''Asymptotic density''' (does not always exist): <math>d(A) : =\lim_{n \to \infty} \frac{a(n)}{n}</math>
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| − | '''Shnirelman's density''': <math>\sigma(A) : =\inf_{n \to \infty} \frac{a(n)}{n}</math>
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| − | '''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>
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| − | <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.
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| − | 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>:
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| − | '''Lower asymptotic density''': <math>_Ld_B(A) : =\liminf_{n \to \infty} \frac{a(n)}{b(n)} </math>
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| − | '''Upper asymptotic density''': <math>_Ud_B(A) : =\limsup_{n \to \infty} \frac{a(n)}{b(n)} </math>
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| − | '''Asymptotic density''' (does not always exist): <math> d_B(A) : =\lim_{n \to \infty}{} \frac{a(n)}{b(n)} </math>
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| − | '''Shnirelman's density''': <math>\sigma_B(A) : =\inf_{n \to \infty} \frac{a(n)}{b(n)} </math>
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| − | '''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>
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| − | Again, the same relations as above hold.
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