hcju/proofwikips huggingface dataset:
Closure of Real Interval is Closed Real Interval
Let $I$ be a non-empty real interval such that one of these holds: : $I = \left({a \,.\,.\, b}\right)$ : $I = \left[{a \,.\,.\, b}\right)$ : $I = \left({a \,.\,.\, b}\right]$ : $I = \left[{a \,.\,.\, b}\right]$ Let $I^-$ denote the closure of $I$. Then $I^-$ is the closed real interval $\left[{a \,.\,.\, b}\right]$.
Topological Product of Compact Spaces
Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Then $T_1 \times T_2$ is compact  both $T_1$ and $T_2$ are compact.
Heine-Cantor Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $M_1$ be compact. Let $f: A_1 \to A_2$ be a continuous mapping. Then $f$ is uniformly continuous.
Continuous Function on Closed Interval is Uniformly Continuous
Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a continuous function. Then $f$ is uniformly continuous on $\closedint a b$.
Sine of Complement equals Cosine
:$\sin \left({\dfrac \pi 2 - \theta}\right) = \cos \theta$
Cosine of Complement equals Sine
:$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$
Continuous Bijection from Compact to Hausdorff is Homeomorphism
Let $T_1$ be a compact space. Let $T_2$ be a Hausdorff space. Let $f: T_1 \to T_2$ be a continuous bijection. Then $f$ is a homeomorphism.
Indirect Proof
Let $P$ be a proposition whose truth value is to be proved (either true or false). There are two aspects to this:
Continuous Mapping of Separation
Let $T$ and $T'$ be topological spaces. Let $A \mid B$ be a separation of $T$. Let $f: T \to T'$ be a mapping such that the restrictions $f \restriction_A$ and $f \restriction_B$ are both continuous. Then $f$ is continuous on the whole of $T$.
Continuity from Union of Restrictions
Let $T_1$ and $T_2$ be topological spaces. Then $f: T_1 \to T_2$ is continuous if either: : $\displaystyle T_1 = \bigcup_{i \mathop = 1}^n V_i$ where: :: each $V_i$ is closed in $T_1$, and  :: the restriction $f \restriction_{V_i}$ is continuous for each $V_i$ or: : $\displaystyle T_1 = \bigcup_{i \mathop \in I} U_i$ where: :: $I$ is any (possibly infinite) indexing set; :: each $U_i$ is open in $T_1$, and :: the restriction $f \restriction_{U_i}$ is continuous for each $U_i$.
Fermat's Last Theorem
:$\forall a, b, c, n \in \Z_{>0}, \; n > 2$, the equation $a^n + b^n = c^n$ has no solutions.
Subset of Real Numbers is Interval iff Connected
Let the real number line $\R$ be considered as a topological space. Let $S$ be a subspace of $\R$. Then $S$ is connected  $S$ is an interval of $\R$. That is, the only subspaces of $\R$ that are connected are intervals.
Continuous Image of Connected Space is Connected
Let $T_1$ and $T_2$ be topological spaces. Let $S_1 \subseteq T_1$ be connected. Let $f: T_1 \to T_2$ be a continuous mapping. Then the image $f \left({S_1}\right)$ is connected.
Union of Connected Sets with Non-Empty Intersections is Connected
Let $T = \struct {S, \tau}$ be a topological space. Let $I$ be an indexing set. Let $\AA = \family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $\AA$ be such that no two of its elements are disjoint: :$\forall B, C \in \AA: B \cap C \ne \O$ Then $\displaystyle \bigcup \AA$ is itself connected.
Finite Product Space is Connected iff Factors are Connected/Basis for the Induction
Let $T_1$ and $T_2$ be topological spaces. Then the product space $T_1 \times T_2$ is connected  $T_1$ and $T_2$ are connected.
Set between Connected Set and Closure is Connected
Let $T$ be a topological space. Let $H$ be a connected set of $T$. Let $H \subseteq K \subseteq H^-$, where $H^-$ denotes the closure of $H$.  Then $K$ is connected.
Path-Connected Space is Connected
Let $T$ be a topological space which is path-connected. Then $T$ is connected.
Hausdorff Paradox
There is a disjoint decomposition of the sphere $\mathbb S^2$ into four sets $A, B, C, D$ such that $A, B, C, B \cup C$ are all congruent and $D$ is countable.
Closed Topologist's Sine Curve is not Path-Connected
Let $G$ be the graph of the function $y = \sin \left({\dfrac 1 x}\right)$ for $x > 0$. Let $J$ be the line segment joining the points $\left({0, -1}\right)$ and $\left({0, 1}\right)$ in $\R^2$. Then while $G \cup J$ is connected, it is '''not''' path-connected.
Sum of Two Sides of Triangle Greater than Third Side
Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side. {{:Euclid:Proposition/I/20}}
Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides
Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle. {{:Euclid:Proposition/I/21}}
Construction of Triangle from Given Lengths
Given three straight lines such that the sum of the lengths of any two of the lines is greater than the length of the third line, it is possible to construct a triangle having the lengths of these lines as its side lengths. {{:Euclid:Proposition/I/22}}
Connected Open Subset of Euclidean Space is Path-Connected
Let $\R^n$ be a Euclidean $n$-space. Let $U$ be a connected open subset of $\R^n$. Then $U$ is path-connected.
Joining Paths makes Another Path
Let $T$ be a topological space. Let $I \subseteq \R$ be the closed real interval $\closedint 0 1$. Let $f, g: I \to T$ be paths in $T$ from $a$ to $b$ and from $b$ to $c$ respectively. Let $h: I \to T$ be the mapping given by: $\map h x = \begin{cases} \map f {2x} & : x \in \closedint 0 {\dfrac 1 2} \\ \map g {2x - 1} & : x \in \closedint {\dfrac 1 2} 1 \end{cases}$ Then $h$ is a path in $T$.
Hinge Theorem
If two triangles have two pairs of sides which are the same length, the triangle with the larger included angle also has the larger third side. {{:Euclid:Proposition/I/24}}
Equivalence of Definitions of Component
{{TFAE|def = Component (Topology)|view = Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$.
Sequentially Compact Metric Subspace is Sequentially Compact in Itself iff Closed
Let $M$ be a metric space. Let $C \subseteq M$ be a subspace of $M$ which is sequentially compact in $M$. Then $C$ is sequentially compact in itself  $C$ is closed in $M$.
Real Number Line is Complete Metric Space
The real number line $\R$ with the usual (Euclidean) metric forms a complete metric space.
Series Law for Extremal Length
Let $X$ be a Riemann surface. Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every $\gamma \in \Gamma$ contain a $\gamma_1 \in \Gamma_1$ and a $\gamma_2 \in \Gamma_2$ such that $\gamma_1 \cap \gamma_2 = \varnothing$. Then the extremal lengths of $\Gamma_1$, $\Gamma_2$ and $\Gamma$ satisfy: :$\lambda \left({\Gamma}\right) \ge \lambda \left({\Gamma_1}\right) + \lambda \left({\Gamma_2}\right)$
Extremal Length of Union
Let $X$ be a Riemann surface. Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.  Then the extremal length of their union satisfies: :$\dfrac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)} \le \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$ Suppose that additionally $\Gamma_1$ and $\Gamma_2$ are disjoint in the following sense: there exist disjoint Borel subsets: :$A_1, A_2 \subseteq X$ such that $\displaystyle \bigcup \Gamma_1 \subset A_1$ and $\displaystyle \bigcup \Gamma_2 \subset A_2$ Then :$\dfrac 1 {\lambda \left({\Gamma_1 \cup \Gamma_2}\right)} = \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$
Comparison Principle for Extremal Length
Let $X$ be a Riemann surface. Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$. Let every element of $\Gamma_1$ contain some element of $\Gamma_2$. Then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by: :$\lambda \left({\Gamma_1}\right) \ge \lambda \left({\Gamma_2}\right)$ More precisely, for every conformal metric $\rho$ as in the definition of extremal length, we have: :$L \left({\Gamma_1, \rho}\right) \ge L \left({\Gamma_2, \rho}\right)$
Reverse Triangle Inequality
Let $M = \struct {X, d}$ be a metric space. Then: :$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$ === Normed Division Ring === {{:Reverse Triangle Inequality/Normed Division Ring}} === Normed Vector Space === {{:Reverse Triangle Inequality/Normed Vector Space}} === Real and Complex Numbers === {{:Reverse Triangle Inequality/Real and Complex Fields}}
Inverse Completion of Integral Domain Exists
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Then an inverse completion of $\struct {D, \circ}$ can be constructed.
Zero of Inverse Completion of Integral Domain
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {K, \circ}$ be the inverse completion of $\struct {D, \circ}$ as defined in Inverse Completion of Integral Domain Exists. Let $x \in K: x = \dfrac p q$ such that $p = 0_D$. Then $x$ is equal to the zero of $K$. That is, ''any'' element of $K$ of the form $\dfrac {0_D} q$ acts as the zero of $K$.
Inverse Completion Less Zero of Integral Domain is Closed
Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\left({K, \circ}\right)$ be the inverse completion of $\left({D, \circ}\right)$. Then $\left({K^*, \circ}\right)$ is closed, where $K^* = K \setminus \left\{{0_K}\right\}$.
Composition of Mappings is Associative
The composition of mappings is an associative binary operation: :$\paren {f_3 \circ f_2} \circ f_1 = f_3 \circ \paren {f_2 \circ f_1}$ where $f_1, f_2, f_3$ are arbitrary mappings which fulfil the conditions for the relevant compositions to be defined.
Group Abelian iff Cross Cancellation Property
Let $G$ be a group. Then the following are equivalent: :$(1): \quad G$ is abelian :$(2): \quad G$ has the cross cancellation property
Group Abelian iff Middle Cancellation Property
Let $G$ be a group. Then the following are equivalent: :$(1): \quad G$ is abelian :$(2): \quad G$ satisfies the middle cancellation property
Rational Numbers form Ordered Field
The set of rational numbers $\Q$ forms an ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$.
Rational Numbers are Close Packed
Let $a, b \in \Q$ such that $a < b$. Then $\exists c \in \Q: a < c < b$. That is, the set of rational numbers is close packed.
Real Addition is Closed
The set of real numbers $\R$ is closed under addition: :$\forall x, y \in \R: x + y \in \R$
Real Multiplication is Well-Defined
The operation of multiplication on the set of real numbers $\R$ is well-defined.
Real Multiplication is Closed
The operation of multiplication on the set of real numbers $\R$ is closed: :$\forall x, y \in \R: x \times y \in \R$
Real Addition is Associative
The operation of addition on the set of real numbers $\R$ is associative: :$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$
Real Multiplication is Associative
The operation of multiplication on the set of real numbers $\R$ is associative: :$\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$
Real Addition is Commutative
The operation of addition on the set of real numbers $\R$ is commutative: :$\forall x, y \in \R: x + y = y + x$
Real Multiplication is Commutative
The operation of multiplication on the set of real numbers $\R$ is commutative: :$\forall x, y \in \R: x \times y = y \times x$
Equidecomposable Nested Sets
Let $A, B, C$ be sets such that $A$ and $C$ are equidecomposable and $A \subseteq B \subseteq C$. Then $B$ and $C$ are equidecomposable.
Rational Numbers form Metric Space
Let $\Q$ be the set of all rational numbers. Let $d: \Q \times \Q \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\Q$ and so $\struct {\Q, d}$ is a metric space.
Rational Numbers form Vector Space
Let $\Q$ be the set of rational numbers. Then the $\Q$-module $\Q^n$ is a vector space. It follows directly, by setting $n = 1$, that the $\Q$-module $\Q$ itself can also be regarded as a vector space.
Five Color Theorem
A planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Limit of (Cosine (X) - 1) over X
:$\displaystyle \lim_{x \mathop \to 0} \frac {\map \cos x - 1} x = 0$
Real Multiplication Distributes over Addition
The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition: :$\forall x, y, z \in \R:$ ::$x \times \paren {y + z} = x \times y + x \times z$ ::$\paren {y + z} \times x = y \times x + z \times x$
Cardano's Formula
Let $P$ be the cubic equation: :$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$ Then $P$ has solutions: :$x_1 = S + T - \dfrac b {3 a}$ :$x_2 = - \dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}$ :$x_3 = - \dfrac {S + T} 2 - \dfrac b {3 a} - \dfrac {i \sqrt 3} 2 \paren {S - T}$ where: :$S = \sqrt [3] {R + \sqrt {Q^3 + R^2} }$ :$T = \sqrt [3] {R - \sqrt {Q^3 + R^2} }$ where: :$Q = \dfrac {3 a c - b^2} {9 a^2}$ :$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$
Ferrari's Method
Let $P$ be the quartic equation: :$a x^4 + b x^3 + c x^2 + d x + e = 0$ such that $a \ne 0$. Then $P$ has solutions: :$x = \dfrac {-p \pm \sqrt {p^2 - 8 q} } 4$ where: :$p = \dfrac b a \pm \sqrt {\dfrac {b^2} {a^2} - \dfrac {4 c} a + 4 y_1}$ :$q = y_1 \mp \sqrt {y_1^2 - \dfrac {4 e} a}$ where $y_1$ is a real solution to the cubic: :$y^3 - \dfrac c a y^2 + \paren {\dfrac {b d} {a^2} - \dfrac {4 e} a} y + \paren {\dfrac {4 c e} {a^2} - \dfrac {b^2 e} {a^3} - \dfrac {d^2} {a^2} } = 0$ '''Ferrari's method''' is a technique for solving this quartic.
Properties of Binomial Coefficients
This page gathers together some of the simpler and more common identities concerning binomial coefficients.
Prime Number Theorem
The prime-counting function $\pi \left({n}\right)$, that is, the number of primes less than $n$, satisfies: :$\displaystyle \lim_{n \mathop \to \infty} \map \pi n \frac {\map \ln n} n = 1$ or equivalently:  :$\map \pi n \sim \dfrac n {\map \ln n}$ where $\sim$ denotes asymptotic equivalence.
Induced Structure Inverse
Let $\struct {G, \oplus}$ be a group whose identity is $e_G$. Let $S$ be a set. Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$. Let $f \in G^S$. Let $f^* \in G^S$ be defined as follows: :$\forall f \in G^S: \forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$ Then $f^*$ is the inverse of $f$ for the pointwise operation induced on $G^S$ by $\oplus$.
Quaternion Group is Hamiltonian
The quaternion group $Q$ is Hamiltonian.
Finite Subset of Metric Space has no Limit Points
Let $M = \left({A, d}\right)$ be a metric space. Let $X \subseteq A$ such that $X$ is finite. Then $X$ has no limit points.
Point in Finite Metric Space is Isolated
Let $M = \struct {A, d}$ be a metric space. Let $X \subseteq A$ such that $X$ is finite. Let $x \in X$. Then $x$ is isolated in $X$.
Point in Finite Hausdorff Space is Isolated
Let $T = \left({S, \tau}\right)$ be a Hausdorff space. Let $X \subseteq S$ such that $X$ is finite. Let $x \in X$. Then $x$ is isolated in $X$.
Subsequence of Sequence in Metric Space with Limit
Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Let $x$ be a limit point of $S = \set {x_n: n \in \N}$, the set of members of $\sequence {x_n}$. Then $\sequence {x_n}$ has a subsequence which converges to $x$.
Primes of form Power Less One
Let $m, n \in \N_{>0}$ be natural numbers. Let $m^n - 1$ be prime. Then $m = 2$ and $n$ is prime.
Lebesgue's Number Lemma
Let $M = \struct {A, d}$ be a metric space. Let $M$ be sequentially compact. Then there exists a Lebesgue number for every open cover of $M$.
Second Subsequence Rule
Let $M = \left({A, d}\right)$ be a metric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$. Suppose $\left \langle {x_n} \right \rangle$ has a subsequence which is unbounded. Then $\left \langle {x_n} \right \rangle$ is divergent.
Open Set Less One Point is Open
Let $M = \left({A, d}\right)$ be a metric space. Let $U \subseteq M$ be an open set of $M$. Let $\alpha \in U$. Then $U \setminus \left\{{\alpha}\right\}$ is open in $M$.
Finite Subspace of Dense-in-itself Metric Space is not Open
Let $M = \struct {A, d}$ be a metric space that is dense-in-itself. Let $U$ be a finite subset of $A$. Then $U$ is not an open set of $M$.
Region Less One Point is Region
Let $M = \left({A, d}\right)$ be a dense-in-itself metric space. Let $R \subseteq M$ be a region of $M$. Let $x \in R$. Then $R \setminus \left\{ {x}\right\}$ is also a region of $M$.
Theorem of Even Perfect Numbers
Let $a \in \N$ be an even perfect number. Then $a$ is in the form: :$2^{n - 1} \paren {2^n - 1}$ where $2^n - 1$ is prime. Similarly, if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect.
Lucas-Lehmer Test
Let $q$ be an odd prime. Let $\sequence {L_n}_{n \mathop \in \N}$ be the recursive sequence in $\Z / \paren {2^q - 1} \Z$ defined by: :$L_0 = 4, L_{n + 1} = L_n^2 - 2 \pmod {2^q - 1}$ Then $2^q - 1$ is prime  $L_{q - 2} = 0 \pmod {2^q - 1}$.
Sum Over Divisors Equals Sum Over Quotients
Let $n$ be a positive integer. Let $f: \Z_{>0} \to \Z_{>0}$ be a mapping on the positive integers. Let $\displaystyle \sum_{d \mathop \divides n} \map f d$ be the sum of $\map f d$ over the divisors of $n$. Then: :$\displaystyle \sum_{d \mathop \divides n} \map f d = \sum_{d \mathop \divides n} \map f {\frac n d}$.
Sum of Reciprocals of Divisors equals Abundancy Index
Let $n$ be a positive integer. Let $\map \sigma n$ denote the sigma function of $n$. Then: :$\displaystyle \sum_{d \mathop \divides n} \frac 1 d = \frac {\map \sigma n} n$ where $\dfrac {\map \sigma n} n$ is the abundancy index of $n$.
Sigma Function of Prime Number
Let $n$ be a positive integer. Let $\map \sigma n$ be the sigma function of $n$. Then $\map \sigma n = n + 1$  $n$ is prime.
Sigma Function of Power of Prime
Let $n = p^k$ be the power of a prime number $p$. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Then: :$\map \sigma n = \dfrac {p^{k + 1} - 1} {p - 1}$
Tau of Power of Prime
Let $n = p^k$ be the power of a prime number $p$. Let $\tau \left({n}\right)$ be the $\tau$ function of $n$. That is, let $\tau \left({n}\right)$ be the number of positive divisors of $n$. Then $\tau \left({n}\right) = k + 1$.
Sum Over Divisors of Multiplicative Function
Let $f: \Z_{>0} \to \Z_{>0}$ be a multiplicative function. Let $n \in \Z_{>0}$. Let $\displaystyle \sum_{d \mathop \divides n} \map f d$ be the sum over the divisors of $n$. Then $\displaystyle \map F n = \sum_{d \mathop \divides n} \map f d$ is also a multiplicative function.
Divisors of Product of Coprime Integers
Let $a \divides b c$, where $b \perp c$. Then $\tuple {r, s}$ satisfying: :$a = r s$, where $r \divides b$ and $s \divides c$ is unique up to absolute value with: :$\size r = \gcd \set {a, b}$ :$\size s = \gcd \set {a, c}$
Unity Function is Completely Multiplicative
Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the constant function: :$\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$ Then $f_1$ is completely multiplicative.
Identity Function is Completely Multiplicative
Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function: :$\forall n \in \Z_{>0}: I_{\Z_{>0}} \left({n}\right) = n$ Then $I_{\Z_{>0}}$ is completely multiplicative.
Tau Function is Multiplicative
The divisor counting function: :$\displaystyle \tau: \Z_{>0} \to \Z_{>0}: \map \tau n = \sum_{d \mathop \divides n} 1$ is multiplicative.
Sigma Function is Multiplicative
The sigma function: :$\displaystyle \sigma: \Z_{>0} \to \Z_{>0}: \sigma \left({n}\right) = \sum_{d \mathop \backslash n} d$ is multiplicative.
Primes of form Power of Two plus One
Let $n \in \N$ be a natural number. Let $2^n + 1$ be prime. Then $n = 2^k$ for some natural number $k$.
Integer has Multiplicative Order Modulo n iff Coprime to n
Let $a$ and $n$ be integers. Let the multiplicative order of $a$ modulo $n$ exist. Then $a \perp n$, that is, $a$ and $n$ are coprime.
Integer to Power of Multiple of Order
Let $a$ and $n$ be integers. Let $a \perp n$, that is, let $a$ and $b$ be coprime. Let $c \in \Z_{>0}$ be the multiplicative order of $a$ modulo $n$. Then $a^k \equiv 1 \pmod n$  $k$ is a multiple of $c$.
Sum of Infinite Geometric Sequence
Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$. Let $z \in S$. Let $\size z < 1$, where $\size z$ denotes: :the absolute value of $z$, for real and rational $z$ :the complex modulus of $z$ for complex $z$. Then $\displaystyle \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$.
Square of Riemann Zeta Function
:$\displaystyle \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map d k} {k^z}$ where: :$\zeta$ is the Riemann zeta function :$d$ is the divisor function.
Derivative of Riemann Zeta Function
The derivative of the Riemann zeta function is: :$\displaystyle \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$
Order of Divisor Function
For all $x \ge 1$: :$\displaystyle \sum_{n \mathop \le x} \map d n = x \log x + \paren {2 \gamma - 1} x + \map \OO {\sqrt x}$ where $\gamma$ is the Euler-Mascheroni constant and $\map d n$ is the divisor function. {{explain|Which particular divisor function? Suggest that it might be $\tau$ function that is being referred to.}}
Order of Möbius Function
Let $\mu$ denote the Möbius function . Then: :$\displaystyle \sum_{n \mathop \le N} \map \mu n = \map o N$ where $o$ denotes little-o notation.
Ingham's Theorem on Convergent Dirichlet Series
Let $\left\vert{a_n}\right\vert \le 1$ {{explain|What exactly is $a_n$ in this context?}} For a complex number $z \in \C$, let $\Re \left({z}\right)$ denote the real part of $z$.  Form the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-z}$ which converges to an analytic function $F \left({z}\right)$ for $\Re \left({z}\right) > 1$. {{explain|We have "$\Re \left({z}\right) > 1$" used here and below to mean $\left\{ {z \in \C: \Re \left({z}\right) > 1}\right\}$? In which case, rather than just call it "$\Re \left({z}\right) > 1$", which is unwieldy and suboptimal, it might be better and clearer to give it a single-symbol identifier.}} Let $F \left({z}\right)$ be analytic throughout $\Re \left({z}\right) \ge 1$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-z}$ converges throughout $\Re \left({z}\right) \ge 1$.
Alexander Polynomial is a Knot Invariant
The Alexander polynomial of an elementary knot $K$ is invariant under Reidemeister moves.
Product Form of Sum on Completely Multiplicative Function
Let $f$ be a completely multiplicative arithmetic function. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \map f n$ be absolutely convergent. Then: :$\displaystyle \sum_{n \mathop = 1}^\infty \map f n = \prod_p \frac 1 {1 - \map f p}$ where the infinite product ranges over the primes.
Euler Phi Function of Product with Prime
Let $p$ be prime and $n \in \Z: n \ge 1$. Then: :$\phi \left({p n}\right) = \begin{cases} \left({p - 1}\right) \phi \left({n}\right) & : p \nmid n \\ p \phi \left({n}\right) & : p \mathrel \backslash n  \end{cases}$ where $\phi \left({n}\right)$ denotes the Euler $\phi$ function of $n$. Thus for all $n \ge 1$ and for any prime $p$, we have that $\phi \left({n}\right)$ divides $\phi \left({p n}\right)$.
Euler Phi Function is Even for Argument greater than 2
Let $n \in \Z: n \ge 1$. Let $\map \phi n$ be the Euler $\phi$ function of $n$. Then $\map \phi n$ is even  $n > 2$.
Cauchy-Goursat Theorem
Let $D$ be a simply connected open subset of the complex plane $\C$. Let $\partial D$ denote the closed contour bounding $D$. Let $f: D \to \C$ be holomorphic everywhere in $D$. Then: :$\displaystyle \oint_{\partial D} \map f z \rd z = 0$