hcju/proofwikips huggingface dataset:
Definition:Repdigit Number
Let $b \in \Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$, be expressed in base $b$. $n$ is a '''repdigit base $b$'''  all of the digits of $n$ are the same digit. For example: :$55555$ or: :$3333_4$ that second one being in base 4.
Definition:Octal Notation
'''Octal''' is another word for base $8$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 8^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7}$
Definition:Well-Ordered Integral Domain
Let $\struct {D, +, \times \le}$ be an ordered integral domain whose zero is $0_D$. === Definition 1 === {{:Definition:Well-Ordered Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Well-Ordered Integral Domain/Definition 2}}
Definition:Gaussian Integral
=== Gaussian Integral of Two Variables === {{:Definition:Gaussian Integral/Two Variables}} === Gaussian Integral of One Variable === {{:Definition:Gaussian Integral/One Variable}}
Definition:Completed Riemann Zeta Function
The '''completed Riemann zeta function''' $\xi: \C \to \C$ is defined on the complex plane $\C$ as: :$\displaystyle \forall s \in \C: \map \xi x := \begin{cases} \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\ \map \xi {1 - s} & : \map \Re s \le 0 \end{cases}$ where $\map \zeta s$ is the Riemann zeta function.
Definition:Order of Entire Function
Let $f: \C \to \C$ be an entire function. Let $\ln$ denote the natural logarithm. === Definition 1 === {{:Definition:Order of Entire Function/Definition 1}} === Definition 2 === {{:Definition:Order of Entire Function/Definition 2}} === Definition 3 === {{:Definition:Order of Entire Function/Definition 3}}
Definition:Proper Subring
A subring $S$ of $R$ is a '''proper subring of $R$'''  $S$ is neither the null ring nor $R$ itself.
Definition:Schwarz Function
Let $f: \R \to \C$ be a function. $f$ is a '''Schwarz function''' : :$\forall c \in \R, n \in \N_0: \left\vert{f^{\left({n}\right)} \left({x}\right)}\right\vert = o \left({\left\vert{x}\right\vert^c}\right)$ where: : $f^{\left({n}\right)}$ denotes the $n$th derivative : $o$ is the little-o notation. {{NamedforDef|Karl Hermann Amandus Schwarz|cat = Schwarz}}
Definition:Common Multiple
Let $S$ be a finite set of non-zero integers, that is: :$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z, x_k \ne 0}$ Let $m \in \Z$ such that all the elements of $S$ divide $m$, that is: :$\forall x \in S: x \divides m$ Then $m$ is a '''common multiple''' of all the elements in $S$. {{expand|Expand the concept to general algebraic expressions, for example that $\paren {2 x + 1} \paren {x - 2}$ is a '''common multiple''' of $\paren {2 x + 1}$ and $\paren {x - 2}$}}
Definition:Ring Automorphism
Let $\struct {R, +, \circ}$ be a ring. Let $\phi: R \to R$ be a (ring) isomorphism. Then $\phi$ is a ring automorphism.
Definition:Even Function
Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ A real function $f: X \to \R$ is an '''even function''' : :$\forall x \in X: \map f {-x} = \map f x$
Definition:Odd Function
Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ A real function $f: X \to \R$ is an '''odd function''' : :$\forall x \in X: \map f {-x} = -\map f x$
Definition:Symmetric Set
Let $G$ be a group. Let $S \subseteq G$ such that: :$\forall x \in S: x^{-1} \in S$ That is, for every element in $S$, its inverse is also in $S$. Then $S$ is a '''symmetric subset of $G$''', or (if $G$ is implicit) $S$ is a '''symmetric set'''. Equivalently, $S \subseteq G$ is a '''symmetric set''' : :$S = S^{-1}$ where $S^{-1}$ is the inverse of $S$. === Symmetric Set of Real Numbers === {{:Definition:Symmetric Set/Real Numbers}}
Definition:Proper Divisor
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $U$ be the group of units of $D$. Let $x, y \in D$. Then $x$ is a '''proper divisor''' of $y$ : :$(1): \quad x \divides y$ :$(2): \quad y \nmid x$ :$(3): \quad x \notin U$ That is: :$(1): \quad x$ is a divisor of $y$ :$(2): \quad x$ is not an associate of $y$ :$(3): \quad x$ is not a unit of $D$ === Integers === As the set of integers form an integral domain, the concept of a '''proper divisor''' is fully applicable to the integers. {{:Definition:Proper Divisor/Integer}}
Definition:Order of Zero
Let $f: \C \to \C$ be a complex function. Let $U \subset \C$ be such that $f$ is analytic in $U$. Let $x \in U$ be a zero of $f$. That is, let $x$ be such that $f \left({x}\right) = 0$. Let $n \in \Z_{\ge 0}$ be the least positive integer such that: : $f^{(n)} \left({x}\right) \ne 0$ where $f^{(n)}$ denotes the $n$th derivative of $f$. Then $n$ is the '''order of the zero''' at $x$. === Simple Zero === {{:Definition:Order of Zero/Simple Zero}}
Definition:Order of Pole
Let $f: \C \to \C$ be a complex function. Let $x \in U \subset \C$ be such that $f$ is analytic in $U \setminus \set x$, with a pole at $x$. By Existence of Laurent Series there is a series: :$\displaystyle \map f z = \sum_{n \mathop \ge n_0}^\infty a_j \paren {z - x}^n$ The '''order''' of the pole at $x$ is defined to be $\size {n_0} > 0$. === Simple Pole === {{:Definition:Order of Pole/Simple Pole}}
Definition:Locally Uniform Convergence/Complex Functions
Let $U \subseteq \C$ be an open set. Let $\sequence {f_n}$ be a be a sequence of functions $f_n : U \to \C$. For $z \in U$, let $\map {D_r} z$ be the disk of radius $r$ about $z$. {{explain|Given that "disk" is just another term for "ball", it remains to be specified whether that ball is open or closed.}} Then $f_n$ '''converges to $f$ locally uniformly''' : :for each $z \in U$, there is an $r > 0$ such that $f_n$ converges uniformly to $f$ on $\map {D_r} z$ and: :$\map {D_r} z \subseteq U$
Definition:Primitive Polynomial (Ring Theory)
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $f \in \Q \sqbrk X$ be such that: :$\cont f = 1$ where $\cont f$ is the content of $f$. That is: :The greatest common divisor of the coefficients of $f$ is equal to $1$. Then $f$ is described as '''primitive'''.
Definition:Vector Sum
Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property. === Component Definition === {{:Definition:Vector Sum/Component Definition}} === Triangle Law === {{:Definition:Vector Sum/Triangle Law}}
Definition:Linearly Dependent
Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Dependent/Sequence}} === Set === {{:Definition:Linearly Dependent/Set}}
Definition:Valuation Ring
Let $\struct {D, +, \circ}$ be a integral domain. Let $K$ be the field of quotients of $D$. Let $K$ be such that: :for all $x \in K$, either $x \in D$ or $x^{-1} \in D$. Then $D$ is a '''valuation ring''' of $K$.
Definition:Tychonoff Topology
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ The '''Tychonoff topology''' on $X$ is defined as the initial topology $\tau$ on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$.
Definition:Fréchet Space (Topology)
{{:Definition:Fréchet Space (Topology)/Definition 1}}
Definition:Regular Space
$\struct {S, \tau}$ is a '''regular space''' : :$\struct {S, \tau}$ is a $T_3$ space :$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.
Definition:Normal Space
$\struct {S, \tau}$ is a '''normal space''' : :$\struct {S, \tau}$ is a $T_4$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
Definition:T3 Space
{{:Definition:T3 Space/Definition 1}}
Definition:T4 Space
{{:Definition:T4 Space/Definition 1}}
Definition:Topologically Distinguishable
Let $T = \left({X, \tau}\right)$ be a topological space. Let $x, y \in X$. Then $x$ and $y$ are '''topologically distinguishable'''  they do not have exactly the same neighborhoods. That is, either: :$\exists U \in \tau: x \in U \subseteq N_x \subseteq X: y \notin N_x$ or: :$\exists V \in \tau: y \in V \subseteq N_y \subseteq X: x \notin N_y$ or both. That is, at least one of the points $x$ and $y$ has a neighborhood that is not a neighborhood of the other. === Topologically Indistinguishable === {{:Definition:Topologically Distinguishable/Indistinguishable}}
Definition:T5 Space
{{:Definition:T5 Space/Definition 1}}
Definition:Completely Normal Space
$\struct {S, \tau}$ is a '''completely normal space''' : :$\struct {S, \tau}$ is a $T_5$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
Definition:Separated by Neighborhoods
Let $T = \left({S, \tau}\right)$ be a topological space. === Sets === {{:Definition:Separated by Neighborhoods/Sets}} === Points === {{:Definition:Separated by Neighborhoods/Points}}
Definition:Urysohn Space
$\struct {S, \tau}$ is an '''Urysohn space''' : :For any distinct elements $x, y \in S$ (that is, $x \ne y$), there exists an Urysohn function for $\set x$ and $\set y$.
Definition:Separated by Closed Neighborhoods
Let $T = \left({S, \tau}\right)$ be a topological space. === Sets === {{:Definition:Separated by Closed Neighborhoods/Sets}} === Points === {{:Definition:Separated by Closed Neighborhoods/Points}}
Definition:Completely Hausdorff Space
{{:Definition:Completely Hausdorff Space/Definition 1}}
Definition:Urysohn Function
Let $T = \struct {S, \tau}$ be a topological space. Let $A, B \subseteq S$ such that $A \cap B = \O$. Let $f: S \to \closedint 0 1$ be a continuous mapping where $\closedint 0 1$ is the closed unit interval. Then $f$ is an '''Urysohn function for $A$ and $B$''' : :$f {\restriction_A} = 0, f {\restriction_B} = 1$ that is: :$\forall a \in A: \map f a = 0$ :$\forall b \in B: \map f b = 1$
Definition:Dyadic Rational
A '''dyadic rational''' is a rational number whose denominator, when expressed in canonical form, is an integral power of $2$. That is, $a \in \Q$ is '''dyadic''' iff: :$\exists p \in \Z, q \in \Z_{>0}: a = \dfrac p {2^q}$
Definition:Tychonoff Space
$\struct {S, \tau}$ is a '''Tychonoff Space''' or '''completely regular space''' : :$\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space :$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.
Definition:Limit Point/Topology/Point
Let $a \in S$. A point $x \in S, x \ne a$ is a '''limit point of $a$'''  every open neighborhood of $x$ contains $a$. That is, it is a limit point of the singleton $\set a$.
Definition:Expansion of Topology
Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1 \subseteq \tau_2$. Then $\tau_2$ is an '''expansion''' of $\tau_1$.
Definition:T3 1/2 Space
$\struct {S, \tau}$ is a '''$T_{3 \frac 1 2}$ space''' : :For any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\set y$.
Definition:Separated by Function
Let $\left({X, \vartheta}\right)$ be a topological space. Let $A, B \subseteq X$. Then $A$ and $B$ are '''separated by function''' iff there exists an Urysohn function for $A$ and $B$. $A$ and $B$ may well be singleton sets $A = \left\{{a}\right\}, B = \left\{{b}\right\}$. In this case $a$ and $b$ are '''separated by function''' iff $A$ and $B$ are '''separated by function'''.  o11tt4onkwpoxop44h4ziff3vtb20rt
Definition:Perfectly T4 Space
$T$ is a '''perfectly $T_4$ space''' : :$(1): \quad T$ is a $T_4$ space :$(2): \quad$ Every closed set in $T$ is a $G_\delta$ set. That is: :Every closed set in $T$ can be written as a countable intersection of open sets of $T$.
Definition:Perfectly Normal Space
$\struct {S, \tau}$ is a '''perfectly normal space''' : :$\struct {S, \tau}$ is a perfectly $T_4$ space :$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
Definition:Semiregular Space
$\struct {S, \tau}$ is a '''semiregular space''' : :$\struct {S, \tau}$ is a Hausdorff ($T_2$) space :The regular open sets of $T$ form a basis for $T$.
Definition:Almost All
=== Measure Space === {{:Definition:Almost Everywhere}} === Set Theory === {{:Definition:Almost All/Set Theory}}
Definition:Finite Intersection Axiom
Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that: :Every set $V_\alpha$ of closed sets of $T$ such that $\displaystyle \bigcap V_\alpha = \O$ contains a finite subset $V_\beta \subseteq V_\alpha$ such that $\displaystyle \bigcap V_\beta = \O$. Then $T$ satisfies the '''finite intersection axiom'''.
Definition:Sigma-Compact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''$\sigma$-compact'''  $S$ is the union of the underlying sets of countably many compact subspaces of $T$.
Definition:Countably Compact Space
=== Definition 1 === {{:Definition:Countably Compact Space/Definition 1}} === Definition 2 === {{:Definition:Countably Compact Space/Definition 2}} === Definition 3 === {{:Definition:Countably Compact Space/Definition 3}} === Definition 4 === {{:Definition:Countably Compact Space/Definition 4}} === Definition 5 === {{:Definition:Countably Compact Space/Definition 5}}
Definition:Countable Finite Intersection Axiom
Let $T = \struct {S, \tau}$ be a topological space. Let $T$ be such that: :Every countable set $V_\alpha$ of closed sets of $T$ such that $\displaystyle \bigcap V_\alpha = \O$ contains a finite subset $V_\beta \subseteq V_\alpha$ such that $\displaystyle \bigcap V_\beta = \O$. Then $T$ satisfies the '''countable finite intersection axiom'''.
Definition:Weakly Countably Compact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''weakly countably compact'''  every infinite subset of $S$ has a limit point in $S$.
Definition:Pseudocompact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''pseudocompact'''  every continuous real-valued function on $S$ is bounded.
Definition:Locally Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. Then $T$ is '''locally compact''' : : every point of $S$ has a neighborhood basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.
Definition:Strongly Locally Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{Definition:Strongly Locally Compact Space/Definition 1}} === Definition 2 === {{Definition:Strongly Locally Compact Space/Definition 2}}
Definition:Convergent Sequence/Topology
{{:Definition:Convergent Sequence/Topology/Definition 1}}
Definition:Convergent Sequence/Analysis
=== Real Numbers === {{:Definition:Convergent Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Convergent Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}}
Definition:Weakly Sigma-Locally Compact Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''weakly $\sigma$-locally compact''' : :$T$ is $\sigma$-compact :$T$ is weakly locally compact.
Definition:Countable Chain Condition
Let $T = \struct {S, \tau}$ be a topological space. $T$ satisfies the '''countable chain condition'''  every disjoint set of open sets of $T$ is countable.
Definition:Refinement of Cover
Let $S$ be a set. Let $\UU = \set {U_\alpha}$ and $\VV = \set {V_\beta}$ be covers of $S$. Then $\VV$ is a '''refinement''' of $\UU$ : :$\forall V_\beta \in \VV: \exists U_\alpha \in \UU: V_\beta \subseteq U_\alpha$ That is,  every element of $\VV$ is the subset of some element of $\UU$. === Finer Cover === {{:Definition:Refinement of Cover/Finer Cover}} === Coarser Cover === {{:Definition:Refinement of Cover/Coarser Cover}}
Definition:Point Finite
Let $S$ be a set. Let $\CC$ be a set of subsets of $S$. Then $\CC$ is '''point finite'''  each element of $S$ is an element of finitely many sets in $\CC$: :$\forall s \in S: \card {\set {C \in \CC: s \in C} } < \infty$
Definition:Locally Finite Cover
Let $T = \struct {S, \tau}$ be a topological space. Let $\CC$ be a cover of $S$. Then $\CC$ is '''locally finite'''  each element of $S$ has a neighborhood which intersects a finite number of sets in $\CC$.
Definition:Star (Topology)
Let $S$ be a set. Let $\CC$ be a cover for $S$. Let $x \in S$. The '''star of $x$ (with respect to $\CC$)''' is defined as: :$\displaystyle x^* := \bigcup \set {U \in \CC: x \in U}$ That is, the union of all sets in $\CC$ which contain $x$.
Definition:Star Refinement
Let $S$ be a set. Let $\CC$ be a cover for $S$. Let $\VV$ be a cover for $S$ such that: :$ \forall x \in S: \exists U \in \CC: x^* \subseteq U$ where $x^*$ is the star of $x$ with respect to $\VV$. That is: :$\displaystyle x^* := \bigcup \set {V \in \VV: x \in V}$ the union of all sets in $\VV$ which contain $x$. Then $\VV$ is a '''star refinement''' of $\CC$.
Definition:Multiplicity (Complex Analysis)
Let $f: \C \to \C$ be a function. Suppose there is $a \in \C$ such that $\map f a = 0$. Then $a$ is said to be a '''zero of multiplicity $k$''' if there exists non-zero $L \in \R$ such that: :$\displaystyle \lim_{z \mathop \to a} \dfrac {\cmod {\map f z} } {\cmod {z - a}^k} = L$ {{definition wanted|Needs to be expanded so as to be relevant to other fields, not just $\C$.}}
Definition:Dirichlet Convolution
Let $f, g$ be arithmetic functions. === Definition 1 === {{:Definition:Dirichlet Convolution/Definition 1}} === Definition 2 === {{:Definition:Dirichlet Convolution/Definition 2}}
Definition:Unit Arithmetic Function
The '''unit arithmetic function''' $u: S \to \Z$ is defined by $u \left({n}\right) = 1$ for all $n$: :$\forall n \in S: u \left({n}\right) = 1$ where $S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$.
Definition:Identity Arithmetic Function
The '''identity arithmetic function''' $\iota: S \to \Z$ is defined for $n \geq 1$ by: :$\forall n \in S: \map \iota n = \delta_{n 1}$ where: :$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$. :$\delta$ is the Kronecker delta. That is: :$\forall n \in S: \map \iota n = \begin {cases} 1 & : n = 1\\ 0 & : n \ne 1 \end {cases}$
Definition:Metacompact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''metacompact'''  every open cover of $S$ has an open refinement which is point finite.
Definition:Paracompact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''paracompact'''  every open cover of $S$ has an open refinement which is locally finite.
Definition:Fully T4 Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''fully $T_4$'''  every open cover of $S$ has a star refinement.
Definition:Fully Normal Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''fully normal''' : :Every open cover of $S$ has a star refinement :All points of $T$ are closed.
Definition:Countably Metacompact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''countably metacompact'''  every countable open cover of $S$ has an open refinement which is point finite.
Definition:Countably Paracompact Space
Let $T = \struct {S, \tau}$ be a topological space. $T$ is '''countably paracompact'''  every countable open cover of $S$ has an open refinement which is locally finite.
Definition:Open Refinement
Let $T = \struct {X, \tau}$ be a topological space. Let $\UU$ and $\VV$ be covers of $X$. Then $\VV$ is an '''open refinement''' of $\UU$ : :$(1): \quad \forall V \in \VV: \exists U \in \UU: V \subseteq U$ :$(2): \quad \VV \subseteq \tau$ That is: :$(1): \quad \VV$ is a refinement of $\UU$ :$(2): \quad$ All elements of $\VV$ are open in $T$
Definition:Little-Omega
Let $f$ and $g$ be real functions. Then: :$f \left({n}\right) \in \omega \left({g \left({n}\right)}\right)$ is equivalent to: :$\displaystyle \lim_{n \to \infty} {\frac{f \left({n}\right)} {g \left({n}\right)}} = \infty$ {{refactor|Extract this into a separate page with proof}} A function $f$ is $\omega \left({g}\right)$ iff $f$ is not $\mathcal O \left({g}\right)$ where $\mathcal O$ is the big-O notation.
Definition:Dirichlet Character
Let $q \in \Z_{>1}$. Let $\paren {\Z / q \Z}$ denote the ring of integers modulo $q$. Let $G = \paren {\Z / q \Z}^\times$ be the group of units of $\paren {\Z / q \Z}$. Let $\C^\times$ be the group of units of $\C$. A '''Dirichlet character modulo q''' is a group homomorphism: :$\chi: G \to \C^\times$ {{refactor|Presumably the following statement is part of the justification for the generalised definition, which needs to be extracted and put into its own page.}} By Reduced Residue System under Multiplication forms Abelian Group, $a + q \Z \in G$  $\map \gcd {a, q} = 1$. It is standard practice to extend $\chi$ to a function on $\Z$ by setting: :$\map \chi A = \begin{cases} \map \chi {a + q \Z} & : \map \gcd {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$ {{explain|The nature of $\chi$ when invoked is not apparent from its notation.  How does one understand, when encountering an instance of $\chi$ whether it means the mapping from $G$ (for some $q$) of just $\Z$?}} === Primitive Character === Let $\chi_0$ be the trivial character modulo $q$. Let $q^*$ be the least divisor of $q$ such that: : $\chi = \chi_0 \chi^*$ where $\chi^*$ is some character modulo $q^*$. If $q = q^*$ then $\chi$ is called '''primitive''', otherwise $\chi$ is '''imprimitive'''.
Definition:Dirichlet L-function
Let $\chi : \left({\Z / q \Z}\right)^\times \to \C^\times$ be a Dirichlet character. {{explain|the notation $\left({\Z / q \Z}\right)^\times$}} A '''Dirichlet $L$-function (associated to $\chi$)''' is a Dirichlet series: :$\displaystyle L \left({s, \chi}\right) = \sum_{n \mathop \ge 1} \chi \left({n}\right) n^{-s}$ for all $s \in \C$ such that the sum converges.
Definition:Dirichlet Density
Let $\mathcal P$ be a set of prime numbers. For $s \in \C$, let $\displaystyle f \left({s}\right) = \sum_{p \mathop \in \mathcal P}\: p^{-s}$. $S$ has '''Dirichlet density''' $\alpha$ : :$\displaystyle \lim_{s \mathop \to 1^+} \left\{{\frac {f \left({s}\right)} {\ln \left({s - 1}\right)}}\right\} = - \alpha$ where $1^+$ indicates a limit from above along the real line. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}}
Definition:Character (Number Theory)
Let $\left({G, +}\right)$ be a finite abelian group. Let $\left({\C_{\ne 0}, \times}\right)$ be the multiplicative group of complex numbers. A '''character''' of $G$ is a group homomorphism: :$\chi: G \to \C_{\ne 0}$
Definition:Completed Dirichlet L-Function
Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$. Let $\displaystyle \kappa = \frac12\left(1 - \chi(-1)\right)$. Let $\delta = 1$ if $\chi$ is the principal character, and $0$ otherwise. The '''completed Dirichlet $L$-function''' for $\chi$ is defined to be :$\displaystyle \Lambda(s,\chi) = \frac{1 + \kappa}{2}\left( s(1-s) \right)^\delta \Gamma\left( \frac{s+\kappa}2 \right)L(s,\chi)$ where $L(s,\chi)$ is the Dirichlet $L$-function for $\chi$.  sg0f5771dwwj52bt574bbvohaja0j5n
Definition:Limit Point/Filter
Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on $S$. A point $x \in S$ is called a '''limit point of $\FF$'''  $\FF$ is finer than the neighborhood filter of $x$.
Definition:Limit Point/Filter Basis
Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on the underlying set $S$ of $T$. Let $\BB$ be a filter basis of $\FF$. === Definition 1 === {{:Definition:Limit Point/Filter Basis/Definition 1}} === Definition 2 === {{:Definition:Limit Point/Filter Basis/Definition 2}}
Definition:Convergent Filter Basis
Let $\struct {S, \tau}$ be a topological space. Let $\BB$ be a filter basis of a filter $\FF$ on $S$. Then $\BB$ '''converges''' to a point $x \in S$ : :$\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$ where $N_x$ is a neighborhood of $x$. That is, a filter basis is '''convergent''' to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.