User:Nomalias

blablabla

polynomials

Chebotarev theorem on roots of_unity generating question 1gq 2gq3gq4

fourier vs contour

+

terms

https://en.wikipedia.org/wiki/Arithmetic_geometry https://en.wikipedia.org/wiki/Arithmetic_dynamics https://en.wikipedia.org/wiki/Arithmetic_topology
https://en.wikipedia.org/wiki/Topological_graph_theory https://en.wikipedia.org/wiki/Shape_theory_(mathematics)

weird

Replica trick++++

snippets

Hilbert 10th problem Pell's conics elliptic curves parallel

	Countable state space	Continuous or general state space
Discrete-time	(discrete-time) Markov chain on a countable or finite state space	Harris chain (Markov chain on a general state space)
Continuous-time	Continuous-time Markov process or Markov jump process	Any continuous stochastic process with the Markov property, e.g., the Wiener process

+

Diophantine equations	Dynamical systems
Rational and integer points on a variety	Rational and integer points in an orbit
Points of finite order on an abelian variety	Preperiodic points of a rational function

+

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https://en.wikipedia.org/wiki/Template:Commutative_local_ring_classes https://en.wikipedia.org/wiki/Template:Commutative_ring_classes https://en.wikipedia.org/wiki/Template:Algebraic_structures https://en.wikipedia.org/wiki/Template:Analogous_fixed-point_theorems

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https://en.wikipedia.org/wiki/Category:Mathematics_navigational_boxes

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stack stuff

$\sum_{k=1}^{n} k^{m}=\sum_{b=1}^{m+1} \binom{n}b\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$+ $\sum_{n=1}^\infty \zeta(2n)x^{2n} = -\frac{\pi x}{2}\cot(\pi x)$ + toric=>rational+ $\sum_{t \in \mathbb{F}_{p^n}} \zeta ^{Tr (t)} = 0$+

stack stuff 2

$d^3\neq a^3+b^3+c^3\Leftrightarrow_{*} d^3\equiv 4,5\pmod9 \Rightarrow d^3\equiv 1,2\pmod3\Leftrightarrow (a+b+c)\equiv 1,2\pmod3$[1]

stuff

$g_n^{-1}(x) = \sup_{i \leq n} \{f_i^{-1}(x)\}=\bigcup_{i=1}^n f_i^{-1}((a,\infty))$+

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$k \mid n\Leftrightarrow p^{k} - 1 \mid p^{n} - 1 \Leftrightarrow x^{p^{k}} - x \mid x^{p^{n}} - x.$ + $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ + $\gcd(m,n)=\prod p_i^{\min(m_i,n_i)}$ $x^2 \equiv a \mbox{ } \mbox{mod} \mbox{ } p\Leftrightarrow a^\frac{p-1}{2} \equiv 1\mbox{ } \mbox{mod} \mbox{ } p$ + $(x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p}$ +

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$\left|\vec a\ \vec b\ \vec c\right|=\vec a \cdot (\vec b \times \vec c)$ + $d\Omega = 4 \pi \left(\frac{d\Sigma}{A}\right) \, (\hat{r} \cdot \hat{n})= 4 \pi \left(\frac{d\Sigma}{4\pi r^2}\right) \, (\hat{r} \cdot \hat{n})$ + $\frac{\mathrm{d}q}{\mathrm{d}t} =\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}\,{\rm d}A\ = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}}\, \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}})$ + $\frac{c+di}{c-di}=\frac{c^2-d^2}{c^2+d^2} + \frac{2cd}{c^2+d^2} i$[2]

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https://en.wikipedia.org/wiki/Zsigmondy's_theorem

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$\text{Cotangent Bundles}\leftrightarrow \text{Pull-backs}\leftrightarrow \text{Differentials}$
$\text{Tangent Bundles}\leftrightarrow \text{Push-forward}\leftrightarrow \text{Tangent Vectors}$ +

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Hausdorff moment problem Stieljes moment problem Hamburger moment problem

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https://en.wikipedia.org/wiki/Highly_composite_number https://en.wikipedia.org/wiki/Van_Eck's_sequence

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https://en.wikipedia.org/wiki/Dirichlet_kernel https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials

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https://en.wikipedia.org/wiki/Frobenius_reciprocity

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https://en.wikipedia.org/wiki/Cyclotomic_polynomial https://proofwiki.org/wiki/Reciprocals_of_Odd_Numbers_adding_to_1

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https://en.wikipedia.org/wiki/Chebyshev_function#The_Riemann_hypothesis https://en.wikipedia.org/wiki/Explicit_formulae_(L-function)#Weil's_Explicit_Formula https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture https://mathoverflow.net/questions/62816/the-guinand-weil-explicit-formula-without-entire-function-theory?rq=1

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https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities

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https://en.wikipedia.org/wiki/Generating_function

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combinatorics

$\dbinom{n}{k}=\frac{1}{2\pi i}\oint_{\left|z\right|=1}\frac{\left(1+z\right)^{n}}{z^{k+1}}dz$ + $B_n = {1 \over e}\sum_{k=0}^\infty \frac{k^n}{k!}.$+ ${j\brace k}=\frac{1}{k!}\sum_{i=0}^{k}\binom{k}{i}\left(-1\right)^{i}\left(k-i\right)^{j}$ +

+++++++++++

$f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.$+ $f^{-(n+1)}(x)= \frac{1}{n!} \int_a^x \left(x-t\right)^n f(t)\,\mathrm{d}t$+ $\oint_{\gamma}(z-c)^{n-k-1}dz=2\pi i\delta_{nk}$+

Universality stuff

https://en.wikipedia.org/wiki/Feigenbaum_constants https://en.wikipedia.org/wiki/Misiurewicz_point https://en.wikipedia.org/wiki/Albert_J._Libchaber#Research https://en.wikipedia.org/wiki/Sharkovskii's_theorem https://en.wikipedia.org/wiki/Kondo_effect https://en.wikipedia.org/wiki/Beta_function_(physics) https://en.wikipedia.org/wiki/Critical_exponent https://en.wikipedia.org/wiki/Universality_class https://en.wikipedia.org/wiki/Kibble-Zurek_mechanism

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https://en.wikipedia.org/wiki/Foias_constant

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Correspondences&dictionaries

https://ncatlab.org/nlab/search?query=correspondence + Schreiber's correspondences+

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Correspondences:
algebraic sets & Ideals
Field subextension & Galois Subgroups
Galois group & Fundemantal group + + +
algebra/geometry & galois/fundamental group Modular forms & Elliptic curves
Automorphic forms & Algebraic curves
Elliptic modular forms & Group representations

Geometric langlands
Kobayashi–Hitchin correspondence
Simpson correspondence
Riemann-Hilbert correspondence
Robinson-Schensted correspondence
Shimura correspondence-Theta correspondence

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Haussdorf-C* algebra duality + + + Lawrence theory Morita's equivalence

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homology-homotopy dictionary+
number field-function field dictionary
Kapranov-Reznikov-Mazur dictionary/arithmetic topology + + + + + arithmetic/knots dictionary +
Diophantine dictionary/Arithmetic dynamics
algebraic geometry dictionary +
wu-yang dictionary
eigensheaf-eigenbrane relation elliptic-parabolic dictionary feynman-intersection number dictionary-like GKPW dictionary+

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votja's conjecture+

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Baez-Stay dictionary

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https://en.wikipedia.org/wiki/Category:Duality_theories

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https://en.wikipedia.org/wiki/Lie_group-Lie_algebra_correspondence#Proof_of_the_homomorphisms_theorem

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Generalizations

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(Milnor conjecture-Thom conjecture) (Witten conjecture-Virasoro conjecture) (K theory-L theory) (Pontryagin_duality-Tannaka-Krein duality+) (Maximun principle-Hopf's Maximum principle) (Padé series-Laurent series-Puiseux series) (Weierstrass factorization-Mittag-Leffer's factorization) (Stone-Weierstrass theorem-Arakelyan's_theorem) (Cantor's_paradox-Ordinal Cantor's paradox(+)) (Russell's_paradox-Girard's_paradox) (Stone's theorem-Stone-von Neumann theorem) (Morse theory-Picard-Lefschetz theory) (Invariant theory-Geometric invariant theory) (https://en.wikipedia.org/wiki/Differential_Galois_theory) (Borel–Weil–Bott theorem-GAGA) (Kloosterman sum-Ramanujan sum) (Weierstrass preparation theoremMalgrange preparation theorem)

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classification miscelanea

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https://en.wikipedia.org/wiki/Uniqueness_theorem

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• https://en.wikipedia.org/wiki/Decomposition_(disambiguation)
• $\Delta \subset \mathbf{C} \subset \widehat{\mathbf{C}}$(complex disk,plane,sphere)
• algebraic classification 2-manifolds: sum connected toris,sum connected projective plane, sphere, geometric classification, topological classification, combinatoric classification
• https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification
• https://en.wikipedia.org/wiki/Wigner's_classification
• https://en.wikipedia.org/wiki/Bianchi_classification
• https://en.wikipedia.org/wiki/Petrov_classification
• https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
• https://en.wikipedia.org/wiki/Enriques–Kodaira_classification
• https://en.wikipedia.org/wiki/Bernstein–Zelevinsky_classification
• https://en.wikipedia.org/wiki/Langlands_classification
• https://en.wikipedia.org/wiki/Segre_classification
• https://en.wikipedia.org/wiki/Crystal_system#Classification_of_lattices
• https://en.wikipedia.org/wiki/Painlevé_transcendents

+ + +

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• https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
• https://en.wikipedia.org/wiki/List_of_finite_simple_groups
• https://en.wikipedia.org/wiki/Compact_group#Classification
• https://en.wikipedia.org/wiki/Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras

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Invariant miscelanea

(Invariant theory-Geometric invariant theory)

• https://en.wikipedia.org/wiki/Classification_theorem
• https://en.wikipedia.org/wiki/Complete_set_of_invariants
• https://en.wikipedia.org/wiki/Invariant_(mathematics)
• (https://en.wikipedia.org/wiki/Invariant_theory)
• https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces
• Eta invariant
• https://en.wikipedia.org/wiki/J-invariant
• https://en.wikipedia.org/wiki/Hasse_invariant

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• https://en.wikipedia.org/wiki/Curvature_invariant
• conformal invariants+

+ +

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https://en.wikipedia.org/wiki/Invariant_differential_operator https://en.wikipedia.org/wiki/Differential_invariant https://en.wikipedia.org/wiki/Differential_operator

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https://en.wikipedia.org/wiki/Quantum_invariant https://en.wikipedia.org/wiki/Periodic_table_of_topological_invariants

Enumerative invariants

Enumerative invariants:
Sympletic category: Donaldson invariants+ Seiberg–Witten invariants Gromov-Witten invariants
FJRW theory Gopakumar-Vafa invariant

duality miscelanea

Reciprocities: https://en.wikipedia.org/wiki/Quadratic_reciprocity http://en.wikipedia.org/wiki/Weil_reciprocity_for_algebraic_curves http://en.wikipedia.org/wiki/Stanley's_reciprocity_theorem +

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+

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https://en.wikipedia.org/wiki/Topology_(electrical_circuits)#Duality

algebraic geo stuff

https://en.wikipedia.org/wiki/Transfer_principle https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

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$\mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}\quad$ $\mu(E)=\frac{dim H^0(X,E)-dim H^1(X,E)}{rank E}+g_X-1$ + + +

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category theory

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five lemmasix operations

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https://en.wikipedia.org/wiki/Universal_property Universal bundle? + https://en.wikipedia.org/wiki/Product_(category_theory) https://en.wikipedia.org/wiki/Exponential_object https://en.wikipedia.org/wiki/Exponential_sheaf_sequence +

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$(Z, \ker(f, g)) ≃ \ker(\text{Hom} (Z, X) ⇉ \text{Hom} (Z, Y ))$ +

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discrete stuff

discrete taylor series discrete taylor series table discrete integration by parts discrete laplacian discrete spectral theory

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stackexchange

The Selberg trace formula is making $PSL(2,\Bbb{R})$ act on $C^\infty(\Gamma \setminus \Bbb{H})$, the Frobenius formula is making G act on $\Bbb{C}[G/H]$ +

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generating functions

Generating functions+$\sum_{n=0}^\infty \frac{s(n)}{n!}x^n = \left(1 - x^2\right)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right).$+connected graph generating function

diferential representations

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$\sum_{n=0}^{\infty} \sum_{k=0}^{n} B(k,n) = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} B(k,n+k)$+https://en.wikipedia.org/wiki/Shift_theorem https://en.wikipedia.org/wiki/Shift_operator https://en.wikipedia.org/wiki/Hasse-Schmidt_derivation

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$e^{D^2} f(x) = \sum_{k=0}^\infty \frac{D^{2k}}{k!}f(x).$

${e^D - 1 \over D}f(x)= \sum_{n=0}^\infty {D^n \over (n+1)!}f(x)$

$e^{D^2}f(x)=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y) e^{-y^2/4}\;dy$

${e^D - 1 \over D}f(x) = \int_x^{x+1} f(u)\,du$

$e^{-D^2}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n H_n(x/2)$

${D \over e^D -1}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n B_n(x)$

+ + +

$\frac{D e^D}{e^D-1}=\sum_{n=0}^\infty \frac{B_n}{n!} D^n=\sum_{n=0}^{\infty} (-1)^{n-1} \frac{\zeta(1-n)}{(n-1)!}D^{n}=-1+\frac{D}{2}-\frac{D^2}{12}+...$

++ +

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$e^{t D}f(x)= f(x+t)$ ++ +

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$(X + D)^n = \sum_{j = 0}^{n }{n\choose j} H_{n-j}(X)D^j$+

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$(1-D)^2X_t = X_t -2X_{t-1} + X_{t-2}\quad$ $(1 - B)^d= \sum_{k=0}^{\infty} \; {d \choose k} \; (-B)^k$ +

Sequences

https://en.wikipedia.org/wiki/Category:Spectral_sequences

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Surgery exact sequence Serre spectral sequence Mapping_class_group

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https://en.wikipedia.org/wiki/Poisson_summation_formula#Derivation

congmatics

https://en.wikipedia.org/wiki/Möbius_transformation

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$Isom^+(\mathbb{H}^3)=PGL(2, \mathbb{C})=PSL(2, \mathbb{C})$+, $Isom(\mathbb{S}^2)=O(3)$+

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$\mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2,SO(3),ISO(\mathbb{R}^2)^+,SL(2,\mathbb{R})=SO(1,3)^+,\pi_1(\mathbb{S}^2)=0,\pi_1(T \cong \R^2/\Z^2)=\Z^2$

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RPⁿ CPⁿ HPⁿ

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$\mathbb T \cong \mbox{U}(1) \cong \mathbb R/\mathbb Z \cong \mbox{SO}(2).$	$e^{i\theta} = \cos\theta + i\sin\theta.$	$\zeta_{SO(2)}(s)=\zeta(s)$
$\mbox{SU}(2)$	$e^{i(\theta/2)(\hat n \cdot \sigma)} = I_2 \cos \theta/2 + i(\hat n \cdot \sigma) \sin \theta/2,$	$\zeta_{SU(2)}(s)=\zeta(s)$
$\mbox{SO}(3)$	$e^{i\theta(\hat n \cdot \mathbf J)} = I_3 + i(\hat n \cdot \mathbf J) \sin \theta + (\hat n \cdot \mathbf J)^2 (\cos \theta - 1),$	$\zeta_{SU(3)}(s)=\sum_{x=1}^{\infty}\sum_{y=1}^{\infty}\frac{1}{(xy(x+y)/2)^s}.$

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Diagramatics

https://commons.wikimedia.org/wiki/Category:Mathematical_diagrams https://commons.wikimedia.org/wiki/Category:Commutative_diagrams https://commons.wikimedia.org/wiki/Category:Group_theory https://commons.wikimedia.org/wiki/File:Projective-representation-lifting.svg +] + + + + + +

matrices stuff

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Khatri rao product

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map	equivalence
$V \to V$	$M \simeq B^{-1} MB$
$V\times V^*\to\mathbb{R}$	$M \simeq B^TMB$

+

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https://en.wikipedia.org/wiki/Matrix_determinant_lemma#See_also

Gaps

https://en.wikipedia.org/wiki/Gap_theorem_(disambiguation) https://en.wikipedia.org/wiki/Spectral_gap_(physics) https://en.wikipedia.org/wiki/Duality_gap https://en.wikipedia.org/wiki/Prime_gap https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture https://en.wikipedia.org/wiki/Yang-Mills_existence_and_mass_gap + [3]

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spectral stuff

https://en.wikipedia.org/wiki/Singular_integral https://en.wikipedia.org/wiki/Singular_trace

linear algebra

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https://en.wikipedia.org/wiki/Matrix_determinant_lemma#See_also + https://en.wikipedia.org/wiki/Matrix_splitting#Matrix_iterative_methods https://en.wikipedia.org/wiki/Matrix_decomposition

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https://en.wikipedia.org/wiki/Matrix_pencil

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https://en.wikipedia.org/wiki/Rouché-Capelli_theorem

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stuff with det,tr

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Poisson summation formulaFrobenius reciprocity Selberg trace formula+++++Poisson=Fourier on circle

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$\frac{d}{dt}e^{X(t)} = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha ~=e^{X}\frac{1 - e^{-\mathrm{ad}_{X}}}{\mathrm{ad}_{X}}\frac{dX(t)}{dt}.$ + + +

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https://en.wikipedia.org/wiki/Trigonometric_functions_of_matrices

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https://en.wikipedia.org/wiki/Nahm_equations#Nahm-Hitchin_description_of_monopoles https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice#Kronig-Penney_model

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http://mathworld.wolfram.com/Convergent.html

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simplex determinant Cayley-Menger_determinant

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$\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\ \|\mathbf{b}\|\cos\theta ,$ $\zeta(n+m)=\sqrt{\zeta(2n)\zeta(2m)}\cos\theta ,$ +

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+

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https://en.wikipedia.org/wiki/Doubly_periodic_function https://en.wikipedia.org/wiki/Fundamental_pair_of_periods

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$\operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} =\operatorname{tr}(\mathbf J(f))$ +
$\Delta f = \nabla^2 f = \nabla \cdot \nabla f =\operatorname{tr}\big(H(f)\big)$ + +

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https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case

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https://en.wikipedia.org/wiki/Vanishing_theorem

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https://en.wikipedia.org/wiki/Analytic_torsion https://en.wikipedia.org/wiki/Crooks_fluctuation_theorem

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$E_n= 2 n \pi k T=\hbar \omega n$ + + +

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$\langle \hat{A}\rangle={1\over Z_0}\operatorname{Tr}\,[\hat{\rho_0}\hat{A}]={1\over Z_0}\sum_n \langle n | \hat{A} |n \rangle e^{-\beta E_n}$

$\hat{\rho_0}=e^{-\beta \hat{H}_0}=\sum_n |n \rangle\langle n |e^{-\beta E_n}$

+ + + + + + +

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$\langle \Gamma_i\Gamma_j \rangle=\operatorname{tr}\{\Gamma_i\Gamma_jR_0\}$ $\langle \Gamma_i(t)\Gamma_j(t') \rangle \propto \delta(t-t')\quad\text{ideally}$ +

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$|\psi(t)\rangle = \exp\left(-\frac{i}{\hbar} \hat H t\right) |q_0\rangle \equiv \exp\left(-\frac{i}{\hbar} \hat H t\right) |0\rangle$

$\left \langle F \bigg| \exp\left( {- {i \over \hbar } \hat H T} \right) \bigg |0 \right \rangle = \int Dq(t) \exp\left[ {i\over \hbar} S \right]$

+

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$\int\exp(-\frac{(X-\operatorname{E}[X])^2}{2\operatorname{E}[(X-\operatorname{E}[X])^2]})=(\operatorname{det}(\operatorname{E}[(X-\operatorname{E}[X])^2))^{\frac{1}{2}}$

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$\Theta \mapsto \operatorname{E}\left[ \, \exp\left( \,i \operatorname{tr}\left(\,\mathbf{X}{\mathbf\Theta}\,\right)\,\right)\, \right] = \left|\, {\mathbf I} - 2i\, {\mathbf\Theta}\,{\mathbf V}\, \right|^{-n/2}$+

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$-\frac{F}{k T} = \ln \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)$
+

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$\operatorname{Index}(D) = \dim\operatorname{Ker}(D)− \dim\operatorname{Ker}(D*)=\operatorname{tr}(\exp(D*D))-\operatorname{tr}(\exp(DD*))$ +
$\frac{1}{Z_{\text{GUE}(n)}} e^{- \frac{n}{2} \mathrm{tr} H^2}$ +

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$\zeta(1-n,a)=-\frac{B_n(a)}{n} \!$ for $n\geq1 \!$+
$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$
$\zeta(-n)=(-1)^n \frac{B_{n+1}}{n+1}$

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https://en.wikipedia.org/wiki/Dedekind_psi_function https://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis

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$p(x) = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} P(t)\, dt = \frac{1}{2\pi} \int_{\mathbf{R}} e^{itx} \overline{\varphi_X(t)}\, dt.$ +

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+ + + $\widehat{f}(x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \widehat\varphi(t)\psi_h(t) e^{-itx} \, dt = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{1}{n} \sum_{j=1}^n e^{it(x_j-x)} \psi(ht) \, dt = \frac{1}{nh} \sum_{j=1}^n \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(ht)\frac{x-x_j}{h}} \psi(ht) \, d(ht) = \frac{1}{nh} \sum_{j=1}^n K\Big(\frac{x-x_j}{h}\Big)$

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$\varphi_{Z_n}(t)=(\varphi_{Y_1}(\frac{t}{\sqrt{n}}))^n=(e^{-\frac{1}{2}(\frac{t}{\sqrt{n}})^2})^n=e^{-\frac{t^2}{2}}$

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$\tilde{K}(p; T)=\tilde{G}_\varepsilon(p)^{T/\varepsilon}=(e^{-\frac{1}{2}(\sqrt{\varepsilon} p)^2})^{T/\varepsilon}= e^{-\frac{T p^2}{2}}$

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$\psi_t(y) = \int \psi_0(x) K(x - y; t) \,dx = \int \psi_0(x) \int_{x(0) = x}^{x(t) = y} e^{iS} \,Dx,$

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$K(x, y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,Dx,$

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$\chi$

$V(G)-E(G)+F(G) = k+1$+ //+ + + + //+ + +

"Index theory"

https://mathoverflow.net/questions/233144/atiyah-singer-theorem-a-big-picture
https://mathoverflow.net/questions/1162/atiyah-singer-index-theorem
http://www.concinnitasproject.org/portfolio/gallery.php?id=Atiyah_Michael

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List of fixed-point theorems +

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$\sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))=|X({\bf F}_q)|$
$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q}))=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index} _x f$
$\sum_i(-1)^i\dim H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv$
+ + + + +

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https://en.wikipedia.org/wiki/Atiyah-Bott_fixed-point_theorem https://en.wikipedia.org/wiki/Supersymmetric_theory_of_stochastic_dynamics#Parisi-Sourlas_approach_to_Langevin_SDEs

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$\sum(-1)^\gamma C^\gamma\,=\chi(M)$
+
$\sum_i \operatorname{index}_{x_i}(v) = \chi(M)\,$ +
$\textrm{Tr}[(-1)^F e^{-\beta H}]=\sum_{p\in\mathbb{Z}}(-1)^pb_p=\chi(M) \ .$ + +

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${\mathcal W} = \operatorname{Tr} (-1)^{\hat n} \langle M_{t't}^* \rangle_\text{noise} = \langle \operatorname{Tr} (-1)^{\hat n} M_{t't}^* \rangle_\text{noise} = I_{L}$. +
$I_L=\operatorname{Tr} (-1)^{\hat n} M_{t't}^*= \sum_{x\in \operatorname{fix} M_{tt'}} \operatorname{sign} \operatorname{det} (\delta_j^i -\partial M_{tt'}^i(x) /\partial x^j)$
$\deg(f,\Omega,p):=\sum_{y\in f^{-1}(p)} \operatorname{sgn} \det(1- Df(y))$ +
$\deg(f) = \sum_{x \in f^{-1}(y)} \operatorname{sgn} (\det(df_x))$+

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$\sigma(n)=\sum_{i\in\mathbb{Z}} (-1)^{i+1}\left(\sigma(n{-}\frac12(3i^2{-}i))+\delta(n,\frac12(3i^2{-}i))\,n\right)=\sigma(n{-}1)+\sigma(n{-}2)-\sigma(n{-}5)-\sigma(n{-}7)+\sigma(n{-}12)+\sigma(n{-}15)+ \cdots$ +

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Maximum modulus principle +Fundamental theorem of algebra
Hairy ball theorem real root

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https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas#Combinatorics_and_topology_method

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Riemann zeta function Hurwitz zeta function Polylogarithm

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delete me? 1+2+3+4+... Riemann+Euler-Maclaurin Darboux's formula

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Analytic torsion Heat kernel signature

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Atiyah Singer index theorem+ Signature operator Equivariant_index theorem

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Bott residue formula

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+ + +

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Supersymmetric atiyah Singer index theorem + +?

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Equivalences:Abel–Plana_formula<=>Euler–Maclaurin_formula<=>Poisson summation formula

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Singular Kernel Regularization

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$[f]=[P_0+\frac{1}{P_1+\frac{1}{P_2+\ddots}}]=\sum_{i \geq 0} (-1)^i[P_i]$ + +

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Algebraic Geometry

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https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces

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https://en.wikipedia.org/wiki/Arithmetic_of_abelian_varieties

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Hilbert's basis theorem Hilbert's Nullstellensatz Hilbert's syzygy theorem

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https://en.wikipedia.org/wiki/Genus-degree_formula https://en.wikipedia.org/wiki/Plücker_formula https://en.wikipedia.org/wiki/Riemann-Hurwitz_formula

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Dual theorems
As the real projective plane, Template:Math, is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:

• Desargues' theorem ⇔ Converse of Desargues' theorem
• Pascal's theorem ⇔ Brianchon's theorem
• Menelaus' theorem ⇔ Ceva's theorem

+

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https://mathoverflow.net/a/17139/142708

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Hodge duality+->Poincaré duality->Grothendieck local duality->Serre duality poincare-serre connection

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https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/Dolbeault_cohomology https://en.wikipedia.org/wiki/Differential_form https://en.wikipedia.org/wiki/Complex_differential_form

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kunneth theorem + + +

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Riemann-Roch Stuff

$\ell (\mathcal K_X - D) = \dim H^0 (X, \omega_X \otimes \mathcal L(D)^\vee),H^0 (X, \omega_X \otimes \mathcal L(D)^\vee)$ Line bundle-Riemann surface Vector Bundle-Complex manifold Quotient stack sheaf-Orbifold Chain-complex sheaf-Scheme Arithmetic+
$\ell(D)-\ell(K-D) = \deg(D) - g + 1=$dimension − correction = degree − genus + 1. +
$\chi(D) = \chi(0) + \frac{1}{2}(D.D - D.K)$ +

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six operationsImages of sheaves

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$e_i f_i= \mathrm{deg}(X/\, X/G)= |G|$+

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https://en.wikipedia.org/wiki/Category:Geometry_of_divisors

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https://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence

Hodge stuff

$\dim H^0(X, \mathbb{C}) - \dim H^1(X, \mathbb{C}) + \dim H^2(X, \mathbb{C}) = 2 \left( \dim H^0(X, \mathcal{O}) - \dim H^1(X, \mathcal{O}) \right)$ +

Homotopy stuff

https://math.stackexchange.com/a/3088/683216

algebraic topology

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$\chi(M \# N) = \chi(M) + \chi(N) - \chi(S^n).$ + $\pi_1(X\times Y)=\pi_1(T)\ast_{\pi_i(S^1)}\pi_1(T)$ +

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$\langle a,b,c,d\ |\ abcd=1\rangle$ $\langle a,b,c\ |\ [a,b]c=1\rangle$ +

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https://en.wikipedia.org/wiki/Homology_(mathematics)

Topological characteristics of closed 1- and 2-manifolds

Manifold		Euler No. χ	Orientability	Betti numbers			Torsion coefficient (1-dimensional)
Symbol	Name			b0	b1	b2
$S^1$	Circle (1-manifold)	0	Orientable	1	1	N/A	N/A
$S^2$	Sphere	2	Orientable	1	0	1	none
$T^2$	Torus	0	Orientable	1	2	1	none
$P^2$	Projective plane	1	Non-orientable	1	0	0	2
$K^2$	Klein bottle	0	Non-orientable	1	1	0	2
	2-holed torus	−2	Orientable	1	4	1	none
	g-holed torus (Genus = g)	2 − 2g	Orientable	1	2g	1	none
	Sphere with c cross-caps	2 − c	Non-orientable	1	c − 1	0	2
	2-Manifold with gTemplate:Nbspholes and cTemplate:Nbspcross-caps (cTemplate:Nbsp>Template:Nbsp0)	2Template:Nbsp−Template:Nbsp(2gTemplate:Nbsp+Template:Nbspc)	Non-orientable	1	(2gTemplate:Nbsp+Template:Nbspc)Template:Nbsp−Template:Nbsp1	0	2

NOTES:

1. For a non-orientable surface, a hole is equivalent to two cross-caps.
2. Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere $S^2$, g = c = 0.

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geometric algebra

https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions

---

https://en.wikipedia.org/wiki/Hodge_star_operator#Derivatives_in_three_dimensions https://en.wikipedia.org/wiki/Exterior_derivative#Exterior_derivative_in_vector_calculus https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms https://en.wikipedia.org/wiki/Exterior_calculus_identities $A \times B = \tfrac{1}{2}(AB - BA)$+ $a \times b = \star (a \wedge b) \,.$+ $\mathbf{a} \times \mathbf{b} = [\mathbf{a}]_{\times} \mathbf{b}$++

sympletic geometry

$\int_M e^{-tH} \omega^n/n! = \sum_p {e^{-tH(p)} \over t^n \prod \alpha_j(p)}.$ + + + !

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Differential geo

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Maxwell's equations-Alternative formulations Mathematical descriptions of the electromagnetic field P-form electrodynamics

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https://en.wikipedia.org/wiki/Category:Theorems_in_Riemannian_geometry

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https://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry
$\underbrace{\operatorname{R}_{ab}}_{\text{Ricci}}\equiv \underbrace{\operatorname{R}^c{}_{acb}}_{\text{Riemann}}= g^{cd} \underbrace{\operatorname{R}_{cadb}}_{\text{Riemann}}$ ++
https://en.wikipedia.org/wiki/Weitzenböck_identity https://en.wikipedia.org/wiki/Bochner's_formula https://en.wikipedia.org/wiki/Bochner_identity

---

$\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \,$ +
$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$ +
+ $\iint_R |N_u \times N_v| \ du\, dv = \iint_R K|X_u \times X_v| \ du\, dv = \iint_S K \ dA$ +
$\operatorname{R}_{ab} = Kg_{ab}. \,$ +
$S=\operatorname{tr}_g \operatorname{Ric}$ + $R_{k \overline{l}}=\partial_{k} \partial_{\overline{l}} \ln (\operatorname{det}(g)),\ \text{Ricci-Chern form}$ + + +
$\frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi};$$exp( - \zeta'(0)) / Vol(S)$ + +

---

$\frac{\partial C}{\partial t}=\frac{\partial^2 C}{\partial s^2} = \kappa n,$	$\int_L \kappa (n) ds + \sum \epsilon_i=2\pi n$
$\frac{\partial S}{\partial t} = D\frac{ \left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} - 2 \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} + \left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2} }{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}} \sqrt{1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2}$	$\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \,$	Minakshisundaram–Pleijel
$\partial_t g_{ij}=-2 R_{ij}.$
	$\int_M \left( |\text{Riem}|^2 - 4 |\text{Ric}|^2 + R^2 \right) d\mu=32\pi^2\chi(M)$	Lovelock $\chi(M) = \left ( \frac{1}{2\pi} \right )^n \int_M \operatorname{Pf}(K)$ +

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• https://en.wikipedia.org/wiki/Wirtinger_derivatives
• https://en.wikipedia.org/wiki/Creation_and_annihilation_operators

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Real n-Cauchy-Riemann: $Df^TDf = (\det(Df))^{2/n}I$ +
$\mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^i db^i = \prod\limits_i da^{\prime i}db^{\prime i}{\det}^{-2}(C^i_j),$ +

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• https://en.wikipedia.org/wiki/Darboux's_theorem
• https://en.wikipedia.org/wiki/Period_mapping

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Volume form + Connection_form torsion form curvature form + spin connection khäler form + Solder form Conformal connection $\int_X \omega \wedge \alpha = \int_X dg \wedge \omega = \int_{\gamma\times (0,\varepsilon)} dg \wedge \omega = \int_{\gamma\times (0,\varepsilon)}d(g\omega) = \int_\gamma \omega.$+

---

$\mathrm{d}p\, \mathrm{d}q = \frac{\partial(p, q)}{\partial(\theta, \varphi)} \mathrm{d}\theta\, \mathrm{d}\varphi = \left(\frac{\partial p}{\partial \theta} \frac{\partial q}{\partial \varphi} - \frac{\partial p}{\partial \varphi} \frac{\partial q}{\partial \theta}\right) \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta \sin \theta\, \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta\, \mathrm{d}\Omega,$ +

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$h = \frac{1}{4} \mathrm{E}\left[(d\log p)^2\right] + \mathrm{E}\left[(d\alpha)^2\right]- \left(\mathrm{E}\left[d\alpha\right]\right)^2- \frac{i}{2}\mathrm{E}\left[d\log p\wedge d\alpha\right] =\frac{1}{4} \mathrm{E}\left[\frac{\partial\log p}{\partial\theta_j}\frac{\partial\log p}{\partial\theta_k}\right]+ \mathrm{E}\left[\frac{\partial\alpha}{\partial\theta_j}\frac{\partial\alpha}{\partial\theta_k}\right]- \mathrm{E}\left[ \frac{\partial\alpha}{\partial\theta_j} \right]\mathrm{E}\left[\frac{\partial\alpha}{\partial\theta_k} \right]- \frac{i}{2}\mathrm{E}\left[\frac{\partial\log p}{\partial\theta_j}\frac{\partial\alpha}{\partial\theta_k}-\frac{\partial\alpha}{\partial\theta_j}\frac{\partial\log p}{\partial\theta_k}\right]$ +

---

$(\nabla \times \mathbf{F})(p)\cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_C \mathbf{F} \cdot d\mathbf{r}\right)$ + ${\frac{\partial g}{\partial \bar{z}}(z_0)}\overset{\mathrm{def}}{=}\lim_{r \to 0}\frac{1}{2\pi i r^2} \oint_{\Gamma(z_0,r)} g(z)\mathrm{d}z,$ +

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Degenerancy theory

covering degenerancy manifold degenerancy

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Poincaré–Hopf theoremHairy ball theorem

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Banach fixed point theorem(existence and uniqueness) Brouwer_fixed-point_theorem(existence) Fixed point degree

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Template:Analogous fixed-point theorems
+ +

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Hall's marrriage theorem equivalences

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Representation stuff

Character theory https://en.wikipedia.org/wiki/Schur_orthogonality_relations https://en.wikipedia.org/wiki/Frobenius-Schur_indicator Weyl character formula Kirillov_character_formula https://en.wikipedia.org/wiki/Dirichlet_character

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--SU(2)&SO(3)-- https://en.wikipedia.org/wiki/3D_rotation_group#Connection_between_SO(3)_and_SU(2) https://en.wikipedia.org/wiki/Representation_theory_of_SU(2) https://en.wikipedia.org/wiki/Representation_of_a_Lie_group#An_example:_The_rotation_group_SO(3) https://en.wikipedia.org/wiki/Spin_spherical_harmonics https://en.wikipedia.org/wiki/Clebsch-Gordan_coefficients

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https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group https://en.wikipedia.org/wiki/Jucys-Murphy_element

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Representation theory Representation of Lie algebra Representation of Lie group Representation of finite groups ℓ-adic representations

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https://en.wikipedia.org/wiki/Representation_theorem https://en.wikipedia.org/wiki/Multiplicity-one_theorem

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conmutative tuff

prime ideals are maximal if nonzero, i.e. dim D≤1->prime ideals are principal->maximal ideals are principal->gcd(a,b)=1⇒(a,b)=1, i.e. coprime then comaximal->D is Bezout->D is a PID +
$l_R(R/I^{[p^e]})=p^{ed}l_R(R/I)$+$\operatorname{reg}(I^r)=rd$+

group stuff

group theory: definitions basics factsnon basic facts

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groups: https://en.wikipedia.org/wiki/Bézout's_identity
Rings: https://en.wikipedia.org/wiki/Chinese_remainder_theorem +

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burside:+

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https://en.wikipedia.org/wiki/Vantieghems_theorem

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$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p,\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p\Leftrightarrow p \text{ prime}$+
$(p-1)! \equiv -1 \pmod p,\Leftrightarrow p \text{ prime}$
$\sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .$+
$\prod_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = (-1)^{n-1} .$+
$\gcd\left(p, \sum_{i=1}^{p-1} i^{p-1}\right)=1\Leftrightarrow p \text{ prime or carmichael}$+

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+ https://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician) https://en.wikipedia.org/wiki/Chebotarev_theorem_on_roots_of_unity [4]

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Cayley's theorem equivalencesWagner-preston theorem +++

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https://en.wikipedia.org/wiki/Deligne-Lusztig_theory

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https://en.wikipedia.org/wiki/Zappa–Szép_product
$H\subset G,K\subset G$=\=>$HK\subset G$+ $H_1\cong K_1,H_2\cong K_2,\ G$=\=>$\frac{H_1}{K_1}\cong\frac{H_1}{K_1}$+ $F_{p}=\langle x\rangle\text{=\=>}F_{p^2}=\langle x\rangle$+ $(\pm1)^2,(\pm12)^2\pmod{143}\equiv 1$+ $G=\langle x\rangle\Rightarrow G/Z(G)=\langle xZ(G)\rangle$+$C_G(C_G(g)) = Z(C_G(g))$+

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$\operatorname{gcd}(|G|,n)||\{x\in G:x^n=1\}|$+

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$|X^{P}|\equiv |X| \mod p\quad \text{(P p-group)}$ +
$a^p \equiv a \mod p\quad \text{(p prime)}$++

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Thompson order formula

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$|G| = |Z(G)| + \sum_{i=1}^r |G:C_G(g_i)|$+ $|S| = |S_0| + \sum_{i=1}^r |G|/|G_i|$+

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$|X/G|=\sum_{x \in X}\frac{1}{|G \cdot x|}$	$|X/G||G|=\sum_{x \in X} |G_x|$	$|X/G||G|=\sum_{g \in G}|X^g|$
$|X|/|G|=\sum_{x \in X/G} \frac{1}{|G_x|}$+	$|X|=\sum_{x \in X/G}|G \cdot x|$	$|X|=|X^G|+\sum_{x \in X/G, |X/G|>1}|G \cdot x|$

$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Stab}(x)|} = \frac{|S|}{|G|},\qquad \sum_{x \in \pi_0(X)} \frac{1}{|\text{Aut}(x)|}$ + + $\sum_{\Lambda}{1\over|{\operatorname{Aut}(\Lambda)}|}= 2\pi^{-n(n+1)/4}\prod_{j=1}^n\Gamma(j/2)\prod_{p\text{ prime}}2m_p(f)$ +
$|G|=|G/G_x||G_x|=|G_x \backslash G||G_x|$+
$|G|=|G/H||H|=|H \backslash G||H|$+
$G/Z(G) \cong Inn(G)$
$Aut(G)/Inn(G) \cong Out(G)$
$X^g = gXg^{-1}$ + + +

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Chebotarev's density theorem[[5] PNTHardy-Ramanujan theorem + + +

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Centralizer-Normalizer Orbit stabilizer coset-index

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https://en.wikipedia.org/wiki/Wedderburn's_little_theorem https://en.wikipedia.org/wiki/Zappa–Szép_product

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Fundamental theorem of abelian groups Fundamental theorem of cyclic groups Fundamental theorem of free groups Jordan–Hölder theorem Finitely generated abelian group Structure theorem for finitely generated modules over a principal ideal domain

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https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Abelian_group#Classification

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mobius table

$\varphi(n) = \sum_{d\mid n} \mu\left( d \right) \cdot \frac{n}{d}$	$n=\sum_{d\mid n}\varphi(d)$
$J_k(n) = \sum_{d|n} \mu\left( d \right) \cdot \frac{n^k}{d}$	$n^k=\sum_{d | n } J_k(d)$
$\Lambda (n) = - \sum_{d \mid n} \mu(d) \log(d) \$	$\log(n) = \sum_{d \mid n} \Lambda(d)$
$\pi_0(x) = \sum_{n=1}^\infty \frac{\mu(n)}n \Pi_0(x^{1/n})$	$\Pi_0(x)=\sum_{n=1}^\infty \frac1n \pi_0(x^{1/n})$
$\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}(x^{1/n})$	$\operatorname{li}(x) = \sum_{n=1}^{\infty} \frac{\operatorname{R}(x^{1/n})}{n}$
$P(s)=\sum_{n>0} \mu(n)\frac{\log\zeta(ns)} n$	$\log\zeta(s)=\sum_{n>0} \frac{P(ns)} n$
$M_k(n)\ =\ \sum\nolimits_{d|n} \mu(\tfrac{n}{d}) N_k(d)$	$N_k(n)\ =\ \sum\nolimits_{d|n} M_k(d)$
$\ln(\Phi_n(x))=\sum_{d\mid n}\mu (\frac{n}{d})\ln(x^d-1)$	$\ln(x^n - 1)=\sum_{d\mid n} \ln(\Phi_d(x))$
$N(q,n)=\frac{1}{n} \sum_{d\mid n} \mu(d)q^{n/d}$	$q^n=\sum_{d\mid n} dN(q,d)$
$\ln(F(x))=\sum_{k>=1}\frac{\mu(x)}{k}log G(x^k)$	$\ln(G(x))=\sum_{k>=1} \frac{F(x^k)}{k}$

+ + + + +

mobius+

$\sum_{d\mid n}\varphi(d)=n$, $\varphi (n) = \sum\limits_{k=1}^n \gcd(k,n) e^{-2\pi i\frac{k}{n}}$; $\sum_{d \mid n} \mu(d)=\delta_{n1}$, $\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,\,n)=1}} e^{2\pi i \frac{k}{n}}$

Lambert

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$\ln(\sum_{n=1}^\infty\frac{1}{n^s}))=\ln((\sum_{n=1}^\infty \frac{\mu(n)}{n^s})^{-1})=\ln(\zeta(s))=\ln(\prod_{p \text{ prime}} (1-p^{-s})^{-1})=\sum_{p,n}\frac{p^{-ns}}{n}=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{n}{n^{s+1}}=...=...$

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$\ln(\sum_{k=0}^\infty a_kq^k)=\ln((\sum_{n=0}^\infty p(k) q^k)^{-1})=\ln(f(q))=\ln(\prod_{k=1}^\infty (1-q^k))=-\sum_{k,n}\frac{q^{nk}}{n}=-\sum_n\frac{\sigma(n)}{n}q^{n}=\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{q^n-1}=,,,$

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$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\mu(k)/k})=-\sum_{k,n}\frac{\mu(k)}{k}\frac{q^{nk}}{n}=-\sum_n\frac{\delta_{n1}}{n}q^{n}=-\sum_{n=1}^\infty \frac{\mu(n)}{n}\,\frac{q^n}{q^n-1}$ + + + + + +
+

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$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\frac{\varphi(k)}{k}})=-\sum_{k,n}\frac{\varphi(k)}{k}\frac{q^{nk}}{n}=-\sum_n \frac{n}{n}q^{n}=-\sum_{n=1}^\infty \frac{\varphi(n)}{n}\,\frac{q^n}{q^n-1}$

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$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\frac{\Lambda(k)}{k}})=-\sum_{k,n}\frac{\Lambda(k)}{k}\frac{q^{nk}}{n}=-\sum_n \frac{\ln(n)}{n}q^{n}=-\sum_{n=1}^\infty \frac{\Lambda(n)}{n}\,\frac{q^n}{q^n-1}$

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$\sum_{n=1}^{\infty} \mathrm{spt}(n) q^n=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty} \frac{q^n \prod_{m=1}^{n-1}(1-q^m)}{1-q^n}$ ++

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$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^{n})=(\sum_{n>0}\tau(n)q^n)^{-24}=(\sum_{n=0}^{\infty}p(n)q^{n-\frac{1}{24}})^{-1}$
$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} = \eta(z)^{24}=\Delta(z),$
+ +

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$\sum_{g=0}^\infty~\sum_{k=1}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})}\text{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}=\sum_{g=0}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})} \text{GW}(g,\beta)q^{\beta}\lambda^{2g-2}$ +

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elliptic stuff

$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$ + $|\mathbb{Z}/p\mathbb{Z}[\alpha]|=|\mathbb{Z}/p\mathbb{Z}|^d$ +
$\int_{\gamma} f(\zeta)\,d\zeta + \int_{\tau^{-1}} f(\zeta) \, d\zeta =\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta = 0.$ +

arithmetic

$[ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty$ ++ $\sigma(\zeta_n) = \zeta_n^{p \mod n}$ +
Finite unramified extension $L/K \Leftrightarrow$ Finite unramified ring extension$\mathcal O_L / \mathcal O_K \Leftrightarrow$ Finite extension $l/k$++ $[ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty$+

primes stuff

$\int_2^Y\left(\sum_{2<p\le x} \log p -\sum_{2<n\le x}1\right)^2\,dx$+$\frac1{\tau_n} \sum_{d\mid n} d^2 - \left(\frac1{\tau_n} \sum_{d\mid n} d\right)^2$+

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Digamma functionWeierstrass zeta function

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$\beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$ + $G \cong G_2 \oplus \bigoplus_{p \, \equiv \, 1 \, (\text{mod } 4)} G_p.$ +

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$\sum_{p\leq x}\frac{1}{p}=\int_2^x \frac{1}{t}\,d(\pi(t))$ +

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$\frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$

$\frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{s}}$

$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$

+

$\Pi_0(x) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^\rho) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t}.$	$\ln \zeta(s) = s\int_0^\infty \Pi_0(x)x^{-s-1}\,\mathrm{d}x.$
$\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho) - \frac{1}{\ln x} + \frac{1}{\pi} \arctan \frac{\pi}{\ln x}$	$\ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x$	$\pi(x) = \operatorname{li}(x) + O\left(\sqrt x \log x\right)$
$\sum_{n \leq x} \Lambda(n) = x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \ln 2\pi - \tfrac{1}{2} \ln (1-x^{-2})=x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \frac{\zeta'(0)}{\zeta(0)} - \sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k}$	$\ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}, \qquad \text{Re}(s) > 1.$

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$\zeta(s) = \dfrac{1}{2(s-1)}(\pi e^{\gamma})^{s/2}\prod_{n=1}^{\infty}(1+\dfrac{s}{2n})e^{-s/2n}\prod_{\rho}(1-\dfrac{s}{\rho})$ +

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++

$\zeta(s)=s \int_0^\infty S(x)\,x^{-s-1}dx=\int_0^\infty S'(x)\,x^{-s}dx$	$\zeta'(s)=-s\int_0^\infty T(x)\,x^{-s-1}dx=-\int_0^\infty T'(x)\,x^{-s}dx$
$\ln\zeta(s)=s\int_0^\infty \Pi_0(x)\,x^{-s-1}dx=\int_0^\infty \Pi_0'(x)\,x^{-s}dx$	$\frac{\zeta'(s)}{\zeta(s)}=-s\int_0^\infty \psi(x)\,x^{-s-1}dx=-\int_0^\infty \psi'(x)\,x^{-s}dx$

$\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx$ + $\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds\quad$ $\frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx$ +

---

$\int_0^\infty x^{s}\ln(1-e^{-x})dx = - \int_0^\infty x^{s-1} \frac{e^{-x}}{1 -e^{-x}} dx$
$\int_{0}^{1} | \mathsf{Li}_{s}(e^{2 \pi i x})|^{2} dx = \sum_{k \geq 1} \left| \frac{e^{2 \pi i k x}}{k^{s}} \right|^{2} = \sum_{k \geq 1} \frac{1}{k^{2s}} = \zeta(2s)$
+

---

$\sum_{p} \frac{\log p}{p^s} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$¿?
$\sum_{p} \frac{\log p}{p^s-1} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$ https://en.wikipedia.org/wiki/Von_Mangoldt_function

---

$\quad \psi'(x)=\ln(x)\,\Pi_0'(x)$ $\quad T'(x)=\ln(x)\,S'(x)$

---

$\quad S[x]=\sum_{n=1}^{\lfloor x\rfloor}1=\lfloor x\rfloor$ $\quad T[x]=\sum_{n=1}^{\lfloor x\rfloor}\log n$

---

$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e ^ x - 1)^{-1}x ^ {s-1}\, \mathrm{d}x=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}\mathrm{d}x\quad$ $\Gamma(s) = \int_0^\infty e^{-x} \,x^{s-1}\, \mathrm{d}x$ +
$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}\int_0^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x,\quad$ $\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$
$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}(\frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(ix)-x^{-\frac12}\right)x^{\frac{s}{2}-1}\,\mathrm{d}x + \frac{1}{2}\int_1^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x)$ +

---

$\gamma = \lim_{n\to\infty}\left(\ln n - \sum_{p\le n}\frac{\ln p}{p-1}\right)= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.$

---

$M = \lim_{n\to\infty}(\sum_{p}^n\frac{1}{p}-\ln(\ln n))$

$\gamma= \lim_{n\to\infty}(\sum_{k=1}^n\frac1{k}-\ln n)$

---

$\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}=-\sum_{p\in\mathcal{P}}\frac{\log(p)}{p^{s}-1}=P_{p-1}'(s)$ +

---

$L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx\quad A(x)=\sum_{n\le x} \chi(n)$
$\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx$+
$\zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} dx$
+ +

---

$\zeta'(s) = -\sum_{n \mathop = 2}^\infty \frac{\ln \left({n}\right)}{n^s}$ +
$\left(\frac{\zeta'(s)}{\zeta(s)}\right)^2 = \sum_{n=1}^\infty \sum_{d|n} \frac{\Lambda(d) \Lambda(n/d)}{ n^{s}}$ +
$\frac{d}{ds}\left(\frac{\zeta^{(k)}(s)}{\zeta(s)}\right)=\frac{\zeta^{(k+1)}(s)}{\zeta(s)}-\frac{\zeta'(s)}{\zeta(s)}\frac{\zeta^{(k)}(s)}{\zeta(s)}$ +

---

$L(s,\chi) = \sum_{n=1} \frac{\chi(n)}{n^s}$	$\Delta(z)= \sum_{n=1}\tau(n)q^n$
$L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}$	$L(s,\tau)=\prod_p\left(1-\frac{\tau(p)}{p^s}+\frac{1}{p^{2s-11}}\right)^{-1}$

---

$\beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$

+

$\omega^p = (x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p},$

+

---

$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\qquad\qquad(1)$ +
$\pi_2(x) \sim 1.32 \frac {x}{(\log x)^2}$ +

---

$p_n\approx n\log n + n(\log \log n - 1),$ $\sum_{n\le x} \sigma_0(n)\approx x\log x + x(2\gamma-1)$
$\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,$ $\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma$ + + + $\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.$++

---

https://en.wikipedia.org/wiki/Prime_gap https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture

---

$\left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}=j(p)-j(q)$ +

---

$\frac{1}{1-x}=\prod_{n\geq 0} (1+x^{2^{n}})$ + +
https://en.m.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant
$\frac{1}{1-x}=\prod_{i=0}(1+x^{1\times b^i}+x^{2\times b^i}+x^{3\times b^i}+...)$

---

$\prod_{n\geq 0}\frac{1}{1-x^{2n+1}}=\prod_{n\geq 0} (1+x^{n})$++

---

$\prod_{i,j} (1-x_i y_j)^{-1}=\sum_\lambda m_\lambda(x) h_{\lambda}(y)=\sum_\lambda s_\lambda(x) s_{\lambda}(y)$ +

---

Rodrigues's formula Li's criterion

---

Szegő_limit_theorems Jensen's formula five value theorem Identity theorem

---

ELSV formula

---

https://en.wikipedia.org/wiki/Fredholm's theorem Fredholm alternative Farkas_lemma Hyperplane separation theorem Hanh Banach separation theorem Positive-definite matrix Positive-definite kernel Positive definiteness

---

---

varieties: Grassman Segre veronese

---

global-local homology global-local homotopy +

---

nowhere differentiable: everywhere continuos, nowhere continuos

---

Moduli stuff

petterson-weil volume+ witten's volume orbifolds volume$\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$+ ++

height stuff

$\max( |A+A|, |A \cdot A| ) \geq c \cdot |A|^{1+\varepsilon}$
$\max(|a|,|b|,|c|) \geq C \cdot rad(abc)^{1+\varepsilon}$+
$\max\{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc))-1.$
$\max\{|x|,|y|,|z|\}<C|\xi-\alpha|^{\frac{-1}{3}}\max(1,\xi^2)$
+

---

https://en.wikipedia.org/wiki/Height_zeta_function

---

https://en.wikipedia.org/wiki/Valuation_(algebra) https://mathoverflow.net/questions/310020/summing-bernoulli-numbers

---

Field	Metric	Completion	zero sequence
Template:Math	Template:Math (usual absolute value)	R	Template:Math
Template:Math	obtained using the p-adic valuation, for a prime number Template:Math	Template:Math [[p-adic number|Template:Math-adic numbers]]	Template:Math
Template:Math (Template:Math any field)	obtained using the Template:Math-adic valuation	Template:Math	Template:Math

+

---

class number

Arithmetic geometry Fermat's squares theorem Minkowski's theorem + Gauss circle problem Dirichlet's divisor problem

---

Class field theory Class number Class number formula
List of number fields with class number one Lists of discriminants of class number 1+ Minkowski's bound
Stark-Heegner_theorem Heegner number Kronecker-Weber_theorem Kummer theory Fundamental discriminant
+ + $\rm\ (x-\alpha)\:(x-\alpha')\ =\ x^2 + x + k\, \text{ Euler-Heegner polynomial } \iff\ \mathbb Z[\alpha]=\mathbb Z[\frac{{1} + \sqrt{1-4k}}{2}]\, \text{ PID}$ + +

---

elliptic and quadratics

+ +

---

+ + + + +

---

+ Template:L-functions-footer Template:Prime number conjectures

---

+ +

---

Perron's formulaShimura correspondence

---

$Q\in E(\Bbb Q)\Rightarrow_{Nagel-Lutz} Q\in E(\Bbb Z)$++

---

$\zeta(s)=\frac{{{\pi^{\frac{s}{2}}}}}{{\Gamma\left({\frac{s}{2}}\right)}}{\int}_0^\infty{{v^{\frac{s}{2}}}\left({\frac{{\theta(iv)-1}}{2}}\right)\frac{{{\rm{d}}v}}{v}}$+
$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$
$\zeta(s)=\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$
$\lim_{s\to 1}(s-1)\zeta(s)=1$

---

$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$
${L_D}(s)\!\!:=\mathop {\prod}\limits_{p\,{\rm{prime}}} {\frac{1}{{1 - {\chi _D}(p){p^{ - s}}}}}$
$\frac{{\sqrt {\left| D \right|} }}{{2\pi }}{L_D}(1)=\frac{{h(D)}}{2}$
$Q(x, y)=Ax^2+Bxy+Cy^2,D=B^{2}-4AC$
$Q(x, y)=Q(ax+by,cx+dy),ad-bc=1$

---

${L_E}(s) = \frac{{{{(2\pi )}^s}}}{{\Gamma (s)}}{\int}_0^\infty {{v^s}{f_E}(iv)\frac{{{\rm{d}}v}}{v}}$
${L_E}(s)\!\!:=\mathop{\prod}\limits_{p\,{\rm{prime}}}{\frac{1}{{1-{a_p}{p^{-s}}+\varepsilon (p){p^{1-2s}}}}}$
${\rm{li}}{{\rm{m}}_{s \to 1}}{(s-1)^{-{r_E}}}{L_E}(s)<\infty$
$\mathop {{\lim }}\limits_{s \to 1}\frac{1}{{{\Omega _E}}}\frac{{{L_E}(s)}}{{{{(s - 1)}^{{r_E}}}}} = {c_E}\left|{{\text{Ш}}(E)}\right|$
${y^2}={x^3}+Ax+B,4A^3+27B^2\neq0$
${(Ncz + d)^{ - 2}}{f_E}\left({\frac{{az + b}}{{Ncz + d}}}\right)={f_E}(z),ad-Nbc=1$

---

$\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}$
$\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1-(N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.$
$\lim_{s \to 1} (s-1) \zeta_K(s) = \frac{2^{r_1} \cdot(2\pi)^{r_2} \cdot \operatorname{Reg}_K \cdot h_K}{w_K \cdot \sqrt{|D_K|}}$

---

$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$ +

---

Winding number-Eisenbud Levine Khimshiashvili signature formula

---

Roth's_theorem-Duffin-Schaeffer conjecture

---

Kutsenov trace formula-Gutzwiller trace formula[6]

---

Min-Max theoremMax-min_inequality

---

Kolmogorov equation-Fokker Planck equation
Koopman operator-Perron-Frobenius operator

---

not recursive function not computable function not ZFC-dependent function bound+

---

tuple primes

+

---

Euler product + +

---

Feller-Tornier constant + Pólya conjecture Chebyshev's bias + + Goldfeld conjecture + Parity_problem

---

$\pi_{n,a}(n) \sim \frac{\pi(n)}{\varphi(n)}$ $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$ + Artin's conjecture +

---

$\Pi^*(x;q,a) = \sum^*_{p\le x, p\equiv a\pmod q} 1 + \sum^*_{p^2\le x, p^2\equiv a\pmod q} \tfrac12 + \sum^*_{p^3\le x, p^3\equiv a\pmod q} \tfrac13 + \cdots = \pi^*(x;q,a) + \tfrac12 \sum_{b\pmod q, b^2\equiv a\pmod q} \pi^*(x^{1/2};q,b) + \tfrac13 \sum_{c\pmod q, c^3\equiv a\pmod q} \pi^*(x^{1/3};q,c) + \cdots$ +

---

$\pi(x) \sim \frac{x}{\ln x} \sim \int_2^x {dt \over \ln t}$	$C_1 = 1$
$\pi_2(x) \sim 2 C_2 \frac{x}{(\ln x)^2} \sim 2 C_2 \int_2^x {dt \over (\ln t)^2}$	$C_2 = \prod_{\textstyle{p\;{\rm prime}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2}\right) \approx 0.660161$
$N(x) \sim C_{LR}\dfrac{x}{\sqrt{\ln(x)}} \sim\text{¿} C_{LR}\int_2^x {dt \over \sqrt{\ln t}}\text{?}$	$C_{LR}=\frac{\pi}{4} \prod_{p \equiv 1\pmod 4} \left(1 - \frac{1}{p^2}\right)^{1/2}=\frac{1}{\sqrt{2}} \prod_{p \equiv 3\pmod 4} \left(1 - \frac{1}{p^2}\right)^{-1/2} \approx 0.764223...$

k-tuple conjecture+ Bateman-Horn conjecture+

---

$\sum_{0<\gamma,\gamma'<T, 0<\gamma-\gamma'< \frac{2\pi\alpha}{\ln T}}1=\frac{T\ln T}{2\pi}(1+o(1))\int_0^\alpha 1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2 \mathrm du.$+ https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture https://en.wikipedia.org/wiki/Sato-Tate_conjecture + +

---

square free distribution +

---

hypergeometric stuff

---

Painlevé's conditionsMalmquist's conditions++

---

Legendre form	hypergeometric differential equation	$j=\frac{256(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2}$
weierstrass form	Picard–Fuchs differential equation	$j=\frac{g_2^3}{g_2^3-27g_3^2}$
Riemann form	Riemann's differential equation

??????????????? Gauss-Manin connection

---

https://en.wikipedia.org/wiki/Confluent_hypergeometric_function

---

---

hipergeometric equation (3 singularities) Heun's equation (4 singularities) Heine-Stieltjes polynomials(n singularities) Schwarz's list

---

• https://en.wikipedia.org/wiki/Modular_elliptic_curve
• https://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions#General_theory
  • https://en.wikipedia.org/wiki/Period_mapping

---

Poincaré series Igusa zeta function

---

Hyperbolic geometryHyperbolic manifold++

---

{5,5}-tilling-Poincaré Sphere+

---

Minimal program model +

---

Abel's theorem converse Abel's theorem

---

Associahedron Pemutohedron

---

https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain

• invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
• primary decomposition + companion matrix yields primary rational canonical form
• primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

---

---

eta stuff

Eta invariant Dirichlet_eta_function Spectral asymmetry Witten index
$\eta(M)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}t^{1/2}\mathrm{Tr}[D\exp(-tD^2)]dt$+
$\frac{1}{4}\left(\eta_{L_{-}}(A)-\eta_{L_{-}}(0)\right)=\frac{c_{2}(G)}{2\pi}I[A]$+

heat table

$\exp(-\frac{cb^2}{2})=\int \frac{dx}{\sqrt{2c\pi}} \exp(-\frac{x^{2}}{2c} \pm ib x)$	$e^{-\frac{1}{2}\langle a | E | a \rangle} \propto \int D\phi \;e^{-\frac{1}{2}\langle \phi | E^{-1} | \phi \rangle} e^{i\langle a | \phi \rangle}$
$\exp(\frac{cb^2}{2})=\int \frac{dx}{\sqrt{2c\pi}} \exp(-\frac{x^{2}}{2c} \pm b x)$	$e^{\frac{1}{2}\langle a | E | a \rangle} \propto \int D\phi \;e^{-\frac{1}{2}\langle \phi | E^{-1} | \phi \rangle} e^{ \langle a | \phi \rangle}$	$e^{D^2}f(x)= \frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty e^{-y^2/4} e^{-yD}f(x)\;dy=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty e^{-y^2/4}f(x-y)\;dy$
$\exp(-\frac{c|b|^2}{2})=\int\frac{dz d\bar{z}}{2i\pi a}\exp(-\frac{|z|^2}{2c}\pm i\overline{b}z\pm i b\overline{z})$
$\exp(\frac{c|b|^2}{2})=\int\frac{dz d\bar{z}}{2i\pi a}\exp(-\frac{|z|^2}{2c}\pm \overline{b}z\pm b\overline{z})$

+ + +

	q	q	q
	$H(Q,P,t)=\sum_{s=0}^\infty \frac{K(P,Q,s)}{s!}(-t)^s \! \quad ?$	$K(P,Q,s)= \frac{1}{\Gamma(s)} \int_0^\infty H(P,Q,t) t^{s-1} dt$	$K \propto \left( \int_V \mathcal D \phi \; e^{- \langle \phi, K\phi\rangle} \right)^{-2}=\det(S)^{-2}=\sum_{n=0}^\infty {(-i)^n\over n!}\left(\prod_{k=1}^n \int_{t_0}^t dt_k\right) \mathcal{T}\left\{\prod_{k=1}^n e^{iH_0 t_k}Ve^{-iH_0 t_k}\right \}.$
	$H(Q,P,t)=\sum_{n=1}^{\infty} f_n(P)f_n(Q) e^{- \lambda_{n}t}$	$K(P,Q,s)=\sum_{n=1}^{\infty} \frac{f_n(P)f_n(Q)}{ \lambda_{n}^s}$
	$\operatorname{tr} H(Q,P,t)=\int_M H(P,P,t)dP=H(s)=\operatorname{tr} e^{-tH} = \sum^\infty_{i=1}e^{-\lambda_i t}$	$\operatorname{tr} K(Q,P,s)=\int_M K(P,P,s)dP=K(s)=\operatorname{tr} K^{-s} = \sum_{\lambda_i} \lambda_i^{-s}=\zeta_K(s)$
		$\det K =e^{-\operatorname{tr} K'(0)}=e^{-(\operatorname{tr} K^{-s})'|_{s=0}}=e^{-(\sum_{\lambda_i} \lambda_i^{-s})'|_{s=0}}=e^{-\sum_{\lambda_i} \ln(\lambda_i) \lambda_i^{-s}|_{s=0}}=e^{-\zeta_K'(0)}$
	q	$\det (I-\lambda K) = \exp (\operatorname{tr} (\ln (I-\lambda K)))=\exp{\left(-\sum_{n=1}^\infty\frac{\operatorname{Tr}(K^n)}{n}\lambda^n\right)}=\sum_{n=0}^\infty (-\lambda)^n \operatorname{Tr } \Lambda^n(K)=\frac{1}{\zeta_K(\lambda)}$
		$K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}\,$
	$H(t)\sim(4\pi t)^{-n/2}\sum^\infty_{m=0}a_m t^m$

+ + + + + + + +

---

$\zeta_{\Delta}(s) = \frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\sum_{\lambda\in\mathrm{Sp}(\Delta)}e^{-\lambda{t}}dt =\frac{1}{\Gamma(2s)}\int_{0}^{\infty}t^{2s-1}\sum_{\lambda\in\mathrm{Sp}(\Delta)}e^{-\sqrt{\lambda}t}dt.$ + $\zeta(k)=\sum_{n=1}^{\infty}\frac{1}{n^{k}}=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}e^{-\pi n^{2}(x_{1}^{2}+\cdots+x_{k}^{2})}dx_{1}\cdots dx_{k}=\sum_{n=1}^{\infty}(\frac{1}{k!}\int_{-\infty}^{\infty}\cdots)^{-k}$?++

---

$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$ +
+

---

$n!=\exp(\ln(n!))=\exp(\sum_n\ln n)=\exp(\sum_n\frac{\ln n}{n^s}|_{s=0})=\exp(-\zeta'(0)))$
$p\#=\exp(\ln(p\#))=\exp(\sum_p\ln p)=\exp(\sum_p\frac{\ln p}{p^s}|_{s=0})=\exp(-P'(0)))$
$p_{\infty}\#=(2\pi)^2$+ $\infty!=\sqrt(2\pi)$+

---

$\ln(n!)\approx n\ln(n)$
+ $\ln(p_n\#)\approx p_n$+

---

$\zeta(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}dx$
$P(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{p>0}e^{-px}dx$

---

$\zeta(s)=\exp(-\sum_p\ln(1-p^{-s}))=\exp(\sum_{p,n}\frac{p^{-ns}}{n})$ +

---

$\phi(q)=\exp(-\sum_k\ln(1-q^k))=\exp(\sum_{k,n}\frac{q^{-nk}}{n})=\exp(\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{q^n-1})$ +

---

$\phi(q)=\exp(-\sum_k\ln(1-q^k))=\exp(\sum_{k,n}\frac{q^{-nk}}{n})=\exp(\sum_{k|m,m}\frac{kq^{-m}}{m})=\exp(\sum_{m}\frac{q^{-m}}{m}\sum_{k|m}k)=\exp(\sum_{m}\frac{q^{-m}}{m}\sigma(m))$
$\log Z(X, T) =\sum_{x \in X}-\log \left(1-T^{\operatorname{deg}(x)}\right)=\sum_{x \in X} \sum_{n=1}^{\infty} \frac{T^{\operatorname{deg}(x) \cdot n}}{n}=\sum_{m=1}^{\infty}\left(\sum_{\operatorname{deg}(x) | m} \operatorname{deg}(x)\right) \frac{T^{m}}{m}=\sum_{m=1}^{\infty}\left|X\left(\mathbb{F}_{q^{m}}\right)\right| \frac{T^{m}}{m}$ + +

---

$Tr_V q^{L_0} = \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}$ +

---

$\wp(z;\Lambda)= -\zeta'(z;\Lambda)=\ln''(\sigma(z;\Lambda)), \mbox{ for any } z \in \Complex$ +
$\frac{1}{2 \pi i} \frac{d}{d z} \log \Delta(z)=1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n z}}{1-e^{2 \pi i n z}}=1-24 \sum_{m=1}^{\infty} \sigma_{1}(m) e^{2 \pi i m z}=1-24\sum_{n>0}\sigma_1(n)q^n=E_{2}(z)$ + +

---

https://en.wikipedia.org/wiki/Hecke_operator https://en.wikipedia.org/wiki/Witt_vector

---

$\sum_{n \leq x} \Lambda(n) =\dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds\quad= x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \ln 2\pi - \tfrac{1}{2} \ln (1-x^{-2})=x - \sum_{\rho} \frac{x^{\rho}}{\rho}- \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2}\sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k}$

---

$\ln \mathcal{L}(\mu,\Sigma) = -{n \over 2} \ln \det(\Sigma) -{1 \over 2} \operatorname{tr} \left[ \Sigma^{-1} \sum_{i=1}^n (x_i-\mu) (x_i-\mu)^\mathrm{T} \right].$

---

$\sum_\rho F(\rho) = \operatorname{Tr}(F(\widehat T )).\!$ + ++++++

---

convergence stuff

+

---

https://en.wikipedia.org/wiki/Operator_topologies

---

https://en.wikipedia.org/wiki/Pasting_lemma

---

Littlewood's three principles of real analysis
Hahn-Banach theorem Open mapping theorem Uniform boundedness principle
Closed graph theorem Bounded inverse theorem zabreikos lemma +
https://en.wikipedia.org/wiki/Template:Functional_analysis
https://en.wikipedia.org/wiki/Category:Theorems_in_functional_analysis

---

Riesz representation theorem Herglotz-Riesz representation Riesz–Markov–Kakutani representation theorem
Spectral theorem

---

https://en.wikipedia.org/wiki/Monotone_convergence_theorem https://en.wikipedia.org/wiki/Dominated_convergence_theorem

---

Katětov–Tong insertion theorem

---

Riesz-Fischer theoremHopf–Rinow theorem

---

https://en.wikipedia.org/wiki/Category:Theorems_in_approximation_theory
https://en.wikipedia.org/wiki/Stone-Weierstrass_theorem https://en.wikipedia.org/wiki/Fejér's_theorem + +
https://en.wikipedia.org/wiki/Runge's_phenomenon https://en.wikipedia.org/wiki/Gibbs_phenomenon

---

https://en.wikipedia.org/wiki/Morera's_theorem https://en.wikipedia.org/wiki/Localization_theorem

---

as=>p=>d+ homeomorphism preserving properties

---

	as	p	d
LLN	T	T	T
LIL	F	T	T
CLT	F	F	T

+

	$\epsilon$	$x$	$f$
Pointwise continuity	x	x	x
Uniform continuity	x		x
equicontinuity	x	x
Uniform equicontinuity	x

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Continuously differentiable $\subseteq$Lipschitz continuous$\subseteq$α-Hölder continuous$\subseteq$uniformly continuous$\subseteq$Continuous function=continuous
$0 < α ≤ 1\Rightarrow$
Lipschitz continuous$\subseteq$absolutely continuous$\subseteq$bounded variation$\subseteq$differentiable$\subseteq$almost everywhere
+

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https://en.wikipedia.org/wiki/Modes_of_convergence_(annotated_index) +

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LLN,LIL,CLT

LIL random matrix

PNT,RH,EK

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$p_n \approx n\log(n)$	$|\gamma_n| \approx \frac{2\pi n }{\ln n}$
$\pi(x)\approx \frac{n}{\log n}$	$N(T)\approx\frac T{2\pi}\ln(\frac T{2\pi\mathrm e})$

+

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+

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https://mathoverflow.net/questions/6889/what-is-the-difference-between-a-zeta-function-and-an-l-function

Geodesics		PGT+		Selberg zeta function	selberg trace formula
P-adic				P-adic L-function
Motives			Motivic zeta function	Motivic L-function
Varietes	Weil Conjectures++		Local zeta function	Hasse-Weil zeta function+
Schemes	Grand RH¿?			arithmetic zeta function
Ideals	ERH			Dedekind zeta function
Characters	GRH			Dirichlet L-function
Naturals	RH	PNT		Riemann zeta	Explicit_formulae

Selberg class Abstract analytic number theory

PNT	RH
$\zeta(1 + it)\neq 0$	$\zeta(1/2 + it)=0$
	$\frac{1}{\sqrt{\frac{1}{2} \log \log T}} \log |\zeta(1/2+it)|\sim N(0,1)$+
$\pi(x) = \operatorname{Li} (x) + o(x)$	$\pi(x) = \operatorname{Li} (x) + O(\sqrt x \log x)$
$\sum_{n \leq x} \mu(n) = o(x)$	$\sum_{n \leq x} \mu(n) = O(x^{1/2+o(1)})$

	Counting	LLN	CLT	Zeros
naturals	primes	PNT		RH
naturals	factors	Hardy ramanujan	Erdős-Kac_theorem
Geodesics	primes	PGT		++
Geodesics	factors		?
Elliptic curves	factors		?

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+

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$\frac{1}{T}\mu(t\le T\,|\,\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}<x)=\lim\limits_{T\to\infty}T^{-1}\int_0^T\mathbf 1_{\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}\leqslant x}\,\mathrm dt=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm e^{-z^2/2}\mathrm dz=\mathbb P(Z\leqslant x)$+ + One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem.
PNT scaled models+ + +
mertens=/=PNT+ +

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CLT

• CLT-frequentist CLT-Bayesian CLT-Arithmetic CLT-Ensemble CLT-Flow CLT

Martingale CLT Functional integralPath integral

• Gibbs entropy-LK entropy-arithmetic entropy?-maximun entropy-openprob entropy
• AEP++LLN+CLT=LAN
• $S=k_{\text{B}}\ln <!--\; \; -->{\mathrm{\Omega <!--\; \Omega \; -->}}_{\mathrm{m}\mathrm{i}\mathrm{c}}=k_{\text{B}}(\ln$