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

$d^{3}\neq a^{3}+b^{3}+c^{3}\Leftrightarrow _{*}d^{3}\equiv 4,5{\pmod {9}}\Rightarrow d^{3}\equiv 1,2{\pmod {3}}\Leftrightarrow (a+b+c)\equiv 1,2{\pmod {3}}$[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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$k\mid n\Leftrightarrow p^{k}-1\mid p^{n}-1\Leftrightarrow x^{p^{k}}-x\mid x^{p^{n}}-x.$ + $\gcd(a^{n}-1,a^{m}-1)=a^{\gcd(n,m)}-1$ + $\gcd(m,n)=\prod p_{i}^{\min(m_{i},n_{i})}$ $x^{2}\equiv a{\mbox{ }}{\mbox{mod}}{\mbox{ }}p\Leftrightarrow a^{\frac {p-1}{2}}\equiv 1{\mbox{ }}{\mbox{mod}}{\mbox{ }}p$ + $(x+yi)^{p}\equiv x^{p}+y^{p}i^{p}\equiv x+(-1)^{\frac {p-1}{2}}yi{\pmod {p}}$ +

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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

${\dbinom {n}{k}}={\frac {1}{2\pi i}}\oint _{\left|z\right|=1}{\frac {\left(1+z\right)^{n}}{z^{k+1}}}dz$ + $B_{n}={1 \over e}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}.$+ Failed to parse (unknown function "\brace"): {\displaystyle {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
Nonabelian Hodge 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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$\mu ({\mathcal {E}})={\frac {deg{\mathcal {E}}}{rk{\mathcal {E}}}}\quad$ $\mu (E)={\frac {dimH^{0}(X,E)-dimH^{1}(X,E)}{rankE}}+g_{X}-1$ + + +

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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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 $PSL(2,{\mathbb {R}})$ act on $C^{\infty }(\Gamma \setminus {\mathbb {H}})$, the Frobenius formula is making G act on ${\mathbb {C}}[G/H]$ +

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

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

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

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

$e^{-D^{2}}\sum _{n=0}^{\infty }a_{n}x^{n}=\sum _{n=0}^{\infty }a_{n}H_{n}(x/2)$

${D \over e^{D}-1}\sum _{n=0}^{\infty }a_{n}x^{n}=\sum _{n=0}^{\infty }a_{n}B_{n}(x)$

+ + +

${\frac {De^{D}}{e^{D}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}D^{n}=\sum _{n=0}^{\infty }(-1)^{n-1}{\frac {\zeta (1-n)}{(n-1)!}}D^{n}=-1+{\frac {D}{2}}-{\frac {D^{2}}{12}}+...$

++ +

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

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

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

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

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

+

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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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$E_{n}=2n\pi kT=\hbar \omega n$ + + +

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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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$\langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}$ $\langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')\quad {\text{ideally}}$ +

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

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Failed to parse (syntax error): {\displaystyle \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 $n\geq 1\!$+
$\zeta (2n)={\frac {(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}}$
$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$

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

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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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$K(x,y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,Dx,$

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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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$\sum _{i}(-1)^{i}{\text{Tr}}({\text{Frob}},H_{c}^{i}(X,{\bf {{Q}_{\ell }))=|X({\bf {F}}_{q})|}}$
$\sum _{i}(-1)^{i}\mathrm {Tr} (f_{*}|H_{k}(X,\mathbb {Q} ))=\sum _{x\in \mathrm {Fix} (f)}\mathrm {index} _{x}f$
$\sum _{i}(-1)^{i}\dim H_{k}(X,\mathbb {Q} )=\sum _{x\in \mathrm {Sing} (v)}\mathrm {index} _{x}v$
+ + + + +

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

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

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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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$\chi (M\#N)=\chi (M)+\chi (N)-\chi (S^{n}).$ + $\pi _{1}(X\times Y)=\pi _{1}(T)\ast _{\pi _{i}(S^{1})}\pi _{1}(T)$ +

---

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

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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.

---

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

---

$\int _{M}K\;dA+\int _{\partial M}k_{g}\;ds=2\pi \chi (M),\,$ +
$\zeta '(\Delta ,0)={\frac {1}{12}}\int _{M}KdA$ +
+ $\iint _{R}|N_{u}\times N_{v}|\ du\,dv=\iint _{R}K|X_{u}\times X_{v}|\ du\,dv=\iint _{S}K\ dA$ +
$\operatorname {R} _{ab}=Kg_{ab}.\,$ +
$S=\operatorname {tr} _{g}\operatorname {Ric}$ + $R_{k{\overline {l}}}=\partial _{k}\partial _{\overline {l}}\ln(\operatorname {det} (g)),\ {\text{Ricci-Chern form}}$ + + +
${\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
$\partial _{t}g_{ij}=-2R_{ij}.$
	$\int _{M}\left(|{\text{Riem}}|^{2}-4|{\text{Ric}}|^{2}+R^{2}\right)d\mu =32\pi ^{2}\chi (M)$	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: $Df^{T}Df=(\det(Df))^{2/n}I$ +
${\mathcal {D}}\psi {\mathcal {D}}{\overline {\psi }}=\prod \limits _{i}da^{i}db^{i}=\prod \limits _{i}da^{\prime i}db^{\prime i}{\det }^{-2}(C_{j}^{i}),$ +

---

• 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 $\int _{X}\omega \wedge \alpha =\int _{X}dg\wedge \omega =\int _{\gamma \times (0,\varepsilon )}dg\wedge \omega =\int _{\gamma \times (0,\varepsilon )}d(g\omega )=\int _{\gamma }\omega .$+

---

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

---

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

---

$(\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {n}} \ {\overset {\underset {\mathrm {def} }{}}{=}}\lim _{A\to 0}\left({\frac {1}{|A|}}\oint _{C}\mathbf {F} \cdot d\mathbf {r} \right)$ + ${{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}{\overset {\mathrm {def} }{=}}\lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,$ +

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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

---

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

---

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
$H\subset G,K\subset G$=\=>$HK\subset G$+ $H_{1}\cong K_{1},H_{2}\cong K_{2},\ G$=\=>${\frac {H_{1}}{K_{1}}}\cong {\frac {H_{1}}{K_{1}}}$+ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{p}=\langle x\rangle\text{=\=>}F_{p^2}=\langle x\rangle} + $(\pm 1)^{2},(\pm 12)^{2}{\pmod {143}}\equiv 1$+ $G=\langle x\rangle \Rightarrow G/Z(G)=\langle xZ(G)\rangle$+$C_{G}(C_{G}(g))=Z(C_{G}(g))$+

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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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$|G|=|Z(G)|+\sum _{i=1}^{r}|G:C_{G}(g_{i})|$+ $|S|=|S_{0}|+\sum _{i=1}^{r}|G|/|G_{i}|$+

---

$|X/G|=\sum _{x\in X}{\frac {1}{|G\cdot x|}}$	$|X/G||G|=\sum _{x\in X}|G_{x}|$	$|X/G||G|=\sum _{g\in G}|X^{g}|$
$|X|/|G|=\sum _{x\in X/G}{\frac {1}{|G_{x}|}}$+	$|X|=\sum _{x\in X/G}|G\cdot x|$	$|X|=|X^{G}|+\sum _{x\in X/G,|X/G|>1}|G\cdot x|$

$\sum _{x\in \pi _{0}(X)}{\frac {1}{|{\text{Stab}}(x)|}}={\frac {|S|}{|G|}},\qquad \sum _{x\in \pi _{0}(X)}{\frac {1}{|{\text{Aut}}(x)|}}$ + + $\sum _{\Lambda }{1 \over |{\operatorname {Aut} (\Lambda )}|}=2\pi ^{-n(n+1)/4}\prod _{j=1}^{n}\Gamma (j/2)\prod _{p{\text{ prime}}}2m_{p}(f)$ +
$|G|=|G/G_{x}||G_{x}|=|G_{x}\backslash G||G_{x}|$+
$|G|=|G/H||H|=|H\backslash G||H|$+
$G/Z(G)\cong Inn(G)$
$Aut(G)/Inn(G)\cong Out(G)$
$X^{g}=gXg^{-1}$ + + +

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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)\$	$\log(n)=\sum _{d\mid n}\Lambda (d)$
$\pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}(x^{1/n})$	$\Pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}(x^{1/n})$
$\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})$	$\operatorname {li} (x)=\sum _{n=1}^{\infty }{\frac {\operatorname {R} (x^{1/n})}{n}}$
$P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n}}$	$\log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n}}$
$M_{k}(n)\ =\ \sum \nolimits _{d|n}\mu ({\tfrac {n}{d}})N_{k}(d)$	$N_{k}(n)\ =\ \sum \nolimits _{d|n}M_{k}(d)$
$\ln(\Phi _{n}(x))=\sum _{d\mid n}\mu ({\frac {n}{d}})\ln(x^{d}-1)$	$\ln(x^{n}-1)=\sum _{d\mid n}\ln(\Phi _{d}(x))$
$N(q,n)={\frac {1}{n}}\sum _{d\mid n}\mu (d)q^{n/d}$	$q^{n}=\sum _{d\mid n}dN(q,d)$
$\ln(F(x))=\sum _{k>=1}{\frac {\mu (x)}{k}}logG(x^{k})$	$\ln(G(x))=\sum _{k>=1}{\frac {F(x^{k})}{k}}$

+ + + + +

mobius+

$\sum _{d\mid n}\varphi (d)=n$, $\varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}$; $\sum _{d\mid n}\mu (d)=\delta _{n1}$, $\mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}}$

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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$\ln(\sum _{k=0}^{\infty }a_{k}q^{k})=\ln((\sum _{n=0}^{\infty }p(k)q^{k})^{-1})=\ln(f(q))=\ln(\prod _{k=1}^{\infty }(1-q^{k}))=-\sum _{k,n}{\frac {q^{nk}}{n}}=-\sum _{n}{\frac {\sigma (n)}{n}}q^{n}=\sum _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{q^{n}-1}}=,,,$

---

$...=...=\ln(g(q))=\ln(\prod _{k\geq 1}(1-q^{k})^{\mu (k)/k})=-\sum _{k,n}{\frac {\mu (k)}{k}}{\frac {q^{nk}}{n}}=-\sum _{n}{\frac {\delta _{n1}}{n}}q^{n}=-\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\,{\frac {q^{n}}{q^{n}-1}}$ + + + + + +
+

---

$...=...=\ln(g(q))=\ln(\prod _{k\geq 1}(1-q^{k})^{\frac {\varphi (k)}{k}})=-\sum _{k,n}{\frac {\varphi (k)}{k}}{\frac {q^{nk}}{n}}=-\sum _{n}{\frac {n}{n}}q^{n}=-\sum _{n=1}^{\infty }{\frac {\varphi (n)}{n}}\,{\frac {q^{n}}{q^{n}-1}}$

---

$...=...=\ln(g(q))=\ln(\prod _{k\geq 1}(1-q^{k})^{\frac {\Lambda (k)}{k}})=-\sum _{k,n}{\frac {\Lambda (k)}{k}}{\frac {q^{nk}}{n}}=-\sum _{n}{\frac {\ln(n)}{n}}q^{n}=-\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n}}\,{\frac {q^{n}}{q^{n}-1}}$

---

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

---

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

---

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

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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

$\int _{2}^{Y}\left(\sum _{2<p\leq x}\log p-\sum _{2<n\leq x}1\right)^{2}\,dx$+${\frac {1}{\tau _{n}}}\sum _{d\mid n}d^{2}-\left({\frac {1}{\tau _{n}}}\sum _{d\mid n}d\right)^{2}$+

---

Digamma functionWeierstrass zeta function

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$\beta (s)=\prod _{p\geq 3 \atop p{\text{ prime}}}{\frac {1}{1-\,\scriptstyle (-1)^{\frac {p-1}{2}}\textstyle p^{-s}}}.$ + $G\cong G_{2}\oplus \bigoplus _{p\,\equiv \,1\,({\text{mod }}4)}G_{p}.$ +

---

$\sum _{p\leq x}{\frac {1}{p}}=\int _{2}^{x}{\frac {1}{t}}\,d(\pi (t))$ +

---

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

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

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

+

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

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

---

++

$\zeta (s)=s\int _{0}^{\infty }S(x)\,x^{-s-1}dx=\int _{0}^{\infty }S'(x)\,x^{-s}dx$	$\zeta '(s)=-s\int _{0}^{\infty }T(x)\,x^{-s-1}dx=-\int _{0}^{\infty }T'(x)\,x^{-s}dx$
$\ln \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)\,x^{-s-1}dx=\int _{0}^{\infty }\Pi _{0}'(x)\,x^{-s}dx$	${\frac {\zeta '(s)}{\zeta (s)}}=-s\int _{0}^{\infty }\psi (x)\,x^{-s-1}dx=-\int _{0}^{\infty }\psi '(x)\,x^{-s}dx$

${\frac {1}{\zeta (s)}}=s\int _{1}^{\infty }{\frac {M(x)}{x^{s+1}}}\,dx$ + $\psi _{0}(x)={\dfrac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }\left(-{\dfrac {\zeta '(s)}{\zeta (s)}}\right){\dfrac {x^{s}}{s}}ds\quad$ ${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-s\int _{1}^{\infty }{\frac {\psi (x)}{x^{s+1}}}\,dx$ +

---

$\int _{0}^{\infty }x^{s}\ln(1-e^{-x})dx=-\int _{0}^{\infty }x^{s-1}{\frac {e^{-x}}{1-e^{-x}}}dx$
$\int _{0}^{1}|{\mathsf {Li}}_{s}(e^{2\pi ix})|^{2}dx=\sum _{k\geq 1}\left|{\frac {e^{2\pi ikx}}{k^{s}}}\right|^{2}=\sum _{k\geq 1}{\frac {1}{k^{2s}}}=\zeta (2s)$
+

---

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

---

$\quad \psi '(x)=\ln(x)\,\Pi _{0}'(x)$ $\quad T'(x)=\ln(x)\,S'(x)$

---

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

---

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }(e^{x}-1)^{-1}x^{s-1}\,\mathrm {d} x={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }x^{s-1}\sum _{n>0}e^{-nx}\mathrm {d} x\quad$ $\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s-1}\,\mathrm {d} x$ +
$\zeta (s)={\frac {1}{2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)}}\int _{0}^{\infty }{\bigl (}\theta (ix)-1{\bigr )}x^{{\frac {s}{2}}-1}\,\mathrm {d} x,\quad$ $\theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }$
$\zeta (s)={\frac {1}{2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)}}({\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (ix)-x^{-{\frac {1}{2}}}\right)x^{{\frac {s}{2}}-1}\,\mathrm {d} x+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (ix)-1{\bigr )}x^{{\frac {s}{2}}-1}\,\mathrm {d} x)$ +

---

$\gamma =\lim _{n\to \infty }\left(\ln n-\sum _{p\leq n}{\frac {\ln p}{p-1}}\right)=\lim _{n\to \infty }\left(-\ln n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx.$

---

$M=\lim _{n\to \infty }(\sum _{p}^{n}{\frac {1}{p}}-\ln(\ln n))$

$\gamma =\lim _{n\to \infty }(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n)$

---

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

---

$L(s,\chi )=s\int _{1}^{\infty }{\frac {A(x)}{x^{s+1}}}\,dx\quad A(x)=\sum _{n\leq x}\chi (n)$
$\zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor x\rfloor }{x^{s+1}}}\,dx$+
$\zeta \left({s}\right)={\frac {s}{s-1}}-s\int _{1}^{\infty }\left\{{x}\right\}x^{-s-1}dx$
+ +

---

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

---

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

---

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

+

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

+

---

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

---

$p_{n}\approx n\log n+n(\log \log n-1),$ $\sum _{n\leq x}\sigma _{0}(n)\approx x\log x+x(2\gamma -1)$
$\lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },$ $\limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma }$ + + + $\lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.$++

---

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
${\frac {1}{1-x}}=\prod _{i=0}(1+x^{1\times b^{i}}+x^{2\times b^{i}}+x^{3\times b^{i}}+...)$

---

$\prod _{n\geq 0}(1+x^{n})=\prod _{n\geq 0}{\frac {1}{1-x^{2n+1}}}$+++
$\prod _{k=1}^{\infty }(1-x^{k})=(\sum _{n=0}^{\infty }p(n)x^{n})^{-1}$

---

$\prod _{i,j}(1-x_{i}y_{j})^{-1}=\sum _{\lambda }m_{\lambda }(x)h_{\lambda }(y)=\sum _{\lambda }s_{\lambda }(x)s_{\lambda }(y)$ +

---

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

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

---

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
+ + Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

---

$Q\in E({\mathbb {Q}})\Rightarrow _{Nagel-Lutz}Q\in E({\mathbb {Z}})$++

---

$\zeta (s)={\frac {\pi ^{\frac {s}{2}}}{\Gamma \left({\frac {s}{2}}\right)}}{\int }_{0}^{\infty }{{v^{\frac {s}{2}}}\left({\frac {\theta (iv)-1}{2}}\right){\frac {{\rm {d}}v}{v}}}$+
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$
$\zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}$
$\lim _{s\to 1}(s-1)\zeta (s)=1$

---

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

---

${L_{E}}(s)={\frac {{(2\pi )}^{s}}{\Gamma (s)}}{\int }_{0}^{\infty }{{v^{s}}{f_{E}}(iv){\frac {{\rm {d}}v}{v}}}$
${L_{E}}(s)\!\!:=\mathop {\prod } \limits _{p\,{\rm {prime}}}{\frac {1}{1-{a_{p}}{p^{-s}}+\varepsilon (p){p^{1-2s}}}}$
${\rm {li}}{{\rm {m}}_{s\to 1}}{(s-1)^{-{r_{E}}}}{L_{E}}(s)<\infty$
$\mathop {\lim } \limits _{s\to 1}{\frac {1}{\Omega _{E}}}{\frac {{L_{E}}(s)}{{(s-1)}^{r_{E}}}}={c_{E}}\left|{{\text{Ш}}(E)}\right|$
${y^{2}}={x^{3}}+Ax+B,4A^{3}+27B^{2}\neq 0$
${(Ncz+d)^{-2}}{f_{E}}\left({\frac {az+b}{Ncz+d}}\right)={f_{E}}(z),ad-Nbc=1$

---

$\zeta _{K}(s)=\sum _{I\subseteq {\mathcal {O}}_{K}}{\frac {1}{(N_{K/\mathbf {Q} }(I))^{s}}}$
$\zeta _{K}(s)=\prod _{P\subseteq {\mathcal {O}}_{K}}{\frac {1}{1-(N_{K/\mathbf {Q} }(P))^{-s}}},{\text{ for Re}}(s)>1.$
$\lim _{s\to 1}(s-1)\zeta _{K}(s)={\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot \operatorname {Reg} _{K}\cdot h_{K}}{w_{K}\cdot {\sqrt {|D_{K}|}}}}$

---

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

---

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 +

---

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

---

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

k-tuple conjecture+ Bateman-Horn conjecture+

---

$\sum _{0<\gamma ,\gamma '<T,0<\gamma -\gamma '<{\frac {2\pi \alpha }{\ln T}}}1={\frac {T\ln T}{2\pi }}(1+o(1))\int _{0}^{\alpha }1-\left({\frac {\sin(\pi u)}{\pi u}}\right)^{2}\mathrm {d} u.$+ 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

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

+ + +

	q	q	q
	$H(Q,P,t)=\sum _{s=0}^{\infty }{\frac {K(P,Q,s)}{s!}}(-t)^{s}\!\quad ?$	$K(P,Q,s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }H(P,Q,t)t^{s-1}dt$	$K\propto \left(\int _{V}{\mathcal {D}}\phi \;e^{-\langle \phi ,K\phi \rangle }\right)^{-2}=\det(S)^{-2}=\sum _{n=0}^{\infty }{(-i)^{n} \over n!}\left(\prod _{k=1}^{n}\int _{t_{0}}^{t}dt_{k}\right){\mathcal {T}}\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k}}Ve^{-iH_{0}t_{k}}\right\}.$
	$H(Q,P,t)=\sum _{n=1}^{\infty }f_{n}(P)f_{n}(Q)e^{-\lambda _{n}t}$	$K(P,Q,s)=\sum _{n=1}^{\infty }{\frac {f_{n}(P)f_{n}(Q)}{\lambda _{n}^{s}}}$
	$\operatorname {tr} H(Q,P,t)=\int _{M}H(P,P,t)dP=H(s)=\operatorname {tr} e^{-tH}=\sum _{i=1}^{\infty }e^{-\lambda _{i}t}$	$\operatorname {tr} K(Q,P,s)=\int _{M}K(P,P,s)dP=K(s)=\operatorname {tr} K^{-s}=\sum _{\lambda _{i}}\lambda _{i}^{-s}=\zeta _{K}(s)$
		$\det K=e^{-\operatorname {tr} K'(0)}=e^{-(\operatorname {tr} K^{-s})'|_{s=0}}=e^{-(\sum _{\lambda _{i}}\lambda _{i}^{-s})'|_{s=0}}=e^{-\sum _{\lambda _{i}}\ln(\lambda _{i})\lambda _{i}^{-s}|_{s=0}}=e^{-\zeta _{K}'(0)}$
	q	$\det(I-\lambda K)=\exp(\operatorname {tr} (\ln(I-\lambda K)))=\exp {\left(-\sum _{n=1}^{\infty }{\frac {\operatorname {Tr} (K^{n})}{n}}\lambda ^{n}\right)}=\sum _{n=0}^{\infty }(-\lambda )^{n}\operatorname {Tr} \Lambda ^{n}(K)={\frac {1}{\zeta _{K}(\lambda )}}$
		$K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-|x-y|^{2}/4t}\,$
	$H(t)\sim (4\pi t)^{-n/2}\sum _{m=0}^{\infty }a_{m}t^{m}$

+ + + + + + + +

---

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

---

$\zeta '(\Delta ,0)={\frac {1}{12}}\int _{M}KdA$ +
+

---

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

---

$\ln(n!)\approx n\ln(n)$
+ $\ln(p_{n}\#)\approx p_{n}$+

---

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

---

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

---

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

---

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

---

$Tr_{V}q^{L_{0}}=\sum _{n\in \mathbf {Z} }\dim V_{n}q^{n}=\prod _{n\geq 1}(1-q^{n})^{-1}$ +

---

$\wp (z;\Lambda )=-\zeta '(z;\Lambda )=\ln ''(\sigma (z;\Lambda )),{\mbox{ for any }}z\in \mathbb {C}$ +
${\frac {1}{2\pi i}}{\frac {d}{dz}}\log \Delta (z)=1-24\sum _{n=1}^{\infty }{\frac {ne^{2\pi inz}}{1-e^{2\pi inz}}}=1-24\sum _{m=1}^{\infty }\sigma _{1}(m)e^{2\pi imz}=1-24\sum _{n>0}\sigma _{1}(n)q^{n}=E_{2}(z)$ + +

---

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

---

Continuously differentiable $\subseteq$Lipschitz continuous$\subseteq$α-Hölder continuous$\subseteq$uniformly continuous$\subseteq$Continuous function=continuous
Failed to parse (syntax error): {\displaystyle 0 < α ≤ 1\Rightarrow}
Lipschitz continuous$\subseteq$absolutely continuous$\subseteq$bounded variation$\subseteq$differentiable$\subseteq$almost everywhere
+

---

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

---

LLN,LIL,CLT

LIL random matrix

PNT,RH,EK

---

$p_{n}\approx n\log(n)$	$|\gamma _{n}|\approx {\frac {2\pi n}{\ln n}}$
$\pi (x)\approx {\frac {n}{\log n}}$	$N(T)\approx {\frac {T}{2\pi }}\ln({\frac {T}{2\pi \mathrm {e} }})$

+

---

+

---

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		?

---

+

---

${\frac {1}{T}}\mu (t\leq T\,|\,\arg(\zeta (1/2+it)/{\sqrt {1/2\log \log t}}<x)=\lim \limits _{T\to \infty }T^{-1}\int _{0}^{T}\mathbf {1} _{\arg(\zeta (1/2+it)/{\sqrt {1/2\log \log t}}\leqslant x}\,\mathrm {d} t={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}\mathrm {e} ^{-z^{2}/2}\mathrm {d} z=\mathbb {P} (Z\leqslant x)$+ + One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem.
PNT scaled models+ + +
mertens=/=PNT+ +

---

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
• $S=k_{\text{B}}\ln \Omega _{\rm {mic}}=k_{\text{B}}(\ln Z_{\rm {can}}+\beta {\bar {E}})=k_{\text{B}}(\ln {\mathcal {Z}}_{\rm {gr}}+\beta ({\bar {E}}-\mu {\bar {N}}))$
  

Hammersley–Clifford theorem

---

https://en.wikipedia.org/wiki/Glivenko-Cantelli_theorem https://en.wikipedia.org/wiki/Donsker's_theorem

---

https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem https://en.wikipedia.org/wiki/Circular_law

---

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

---

---

numbers, fields, curves, p-adic

---

+ +

---

$\lim _{s\to 1}(s-1)\zeta _{K}(s)={\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot \operatorname {Reg} _{K}\cdot h_{K}}{w_{K}\cdot {\sqrt {|D_{K}|}}}}$ + $\lim _{s\to 1}(s-1)\zeta _{K}(s)={\frac {(q-1)^{r_{1}-1}\cdot \operatorname {Reg} _{K}\cdot h_{K}}{\ln(q)\cdot {\sqrt {|D_{K}|}}}}$ +

---

$\zeta _{K}(s)=\zeta (s)\prod _{\rho \neq 1}L(s,\rho )=\color {red}{\zeta (s)\prod _{\pi \neq 1}L(s,\pi )}$ +
$\Phi (G):=\det(X_{gh^{-1}})=\prod _{\chi \in {\widehat {G}}}\left[\sum _{g\in G}\chi (g)X_{g}\right]=\prod _{\rho ~{\rm {irred}}}\det \left(\sum _{g\in G}X_{g}\rho (g)\right)^{\deg \rho }$ +
$\det \left(\sum _{n=1}^{\infty }\Phi (n)n^{-s}\right)=\prod _{j=1}^{\phi (k)}L(s,{\tilde {\chi }}_{j})=\zeta _{\mathbb {Q} (\zeta _{k})}(s)=\sum _{I\subset {\mathcal {O}}_{\mathbb {Q} (\zeta _{k})}}N(I)^{-s}$+

---

$\zeta _{[t=N({\mathfrak {p}})^{-s}]}L(s)=\prod _{{\mathfrak {p}}\in K}{\frac {1}{\det \left[I-N({\mathfrak {p}})^{-s}\rho (\mathbf {Frob} _{\mathfrak {p}}){|V_{{\mathfrak {p}},\rho }}\right]}}=\prod _{n}{\frac {1}{(1-N({\mathfrak {p}})^{-sf})^{\frac {n}{f}}}}$ +
Failed to parse (unknown function "\substack"): {\displaystyle \psi(x,\rho)=\sum_{\substack{{\frak p}\textrm{ unram} \\ N({\frak p})^m\le x}}{\rm tr}\,\rho(\tau({\frak p}))\log N({\frak p})=\sum_{\substack{{\frak p}\textrm{ unram} \\ N({\frak p})^m\le x}}χ(g)\log N({\frak p})} ++
Failed to parse (unknown function "\large"): {\displaystyle L^{\large\rm{Artin}}_{E/K}(\sigma,s)=L^{\large\rm{Hecke}}_{K}(\chi,s)} +

---

$Z(X,t)=\prod _{i=0}^{2\dim X}\det {\big (}1-t{\mbox{Frob}}_{q}|H_{c}^{i}({\overline {X}},{\mathbb {Q} }_{\ell }){\big )}^{(-1)^{i+1}}.$
$Z(X,t)=\prod \ (1-t^{\deg(x)})^{-1}.$ +

---

$\zeta _{\mathbf {A} ^{n}(X)}(s)=\zeta _{X}(s-n)$$\zeta _{\mathbf {P} ^{n}(X)}(s)=\prod _{i=0}^{n}\zeta _{X}(s-i)$ +
$Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}$$Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}$ +

---

$\zeta _{\mathbb {F} _{p}[p^{-s}]}(s)={\frac {1}{1-p^{1-s}}}=\sum _{n}p^{n(1-s)}=\sum _{g\in A{\text{ monic}}}\vert g\vert ^{-s}=\prod _{f{\text{ monic irreducible}}}(\sum _{k}|g|^{-sk})=\prod _{f{\text{ monic irreducible}}}(1-\vert f\vert ^{-s})^{-1}=\prod _{f}{\frac {1}{1-|\mathbb {F} _{p}[x]/(f)|^{-s}}}$++ $\sum q^{-\deg \ f(x)}=\sum f^{-1}=\infty$+

---

$\zeta _{\mathbb {F} _{p}[x]}(s)={\frac {1}{1-px}}=\sum _{n}(px)^{n}=\sum _{d(g)}x^{d(g)}=\prod _{d(f)}(\sum _{k}x^{kd(f)})=\prod _{d(f)}(1-x^{d(f)})^{-1}=\prod _{d}(1-x^{d})^{-\sigma (d)},\quad (\prod _{d(f)}{\frac {1}{1-x^{d(f)}}}=\prod _{d}\prod _{d(f)=d}{\frac {1}{1-x^{d}}}=\prod _{d}{\frac {1}{(1-x^{d})^{\sum _{d(f)=d}1}}}\prod _{d}{\frac {1}{(1-x^{d})^{\sigma (d)}}})$ + ++ +

---

$\zeta _{R}(s)=\prod _{m\subset R}{\frac {1}{1-|R/m|^{-s}}}$ [+]

---

Failed to parse (unknown function "\Large"): {\displaystyle \zeta_G(s)=\sum_{H\le_{\Large f} G}[G:H]^{-s}=\sum_{n\ge1}\sigma(G)\,n^{-s}} +

---

$\sum _{f}\sigma (f)|f|^{-s}=(1-q^{1-s})^{-1}(1-q^{2-s})^{-1}$+ $\sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)$+ $Z(t)={\frac {P(t)}{(1-t)(1-qt)}}\$+ $Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}}$+

---

$\prod _{n\geq 0}{\frac {1}{1-x^{2n+1}}}=\prod _{n\geq 0}(1+x^{n})$++ $\sum _{n=0}^{\infty }[({\mathbb {A} }^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L} }^{m+1}t^{m}}}$ +

---

(Elliptic function-Elliptic curve) (Modular form-Modular curve)

---

function field analogy+ + + +

Riemann-Roch_theorem	arithmetic Riemann-Roch theorem++
Hodge conjecture	aritmetic Hodge conjecture

Elliptic curve	Elliptic surface
Faltings's theorem	Bombieri-Lang conjecture

PNT	Landau PNT+	PNT function field
RH	Dedekind zeta function	Goss zeta function
Dirichlet arithmetic progression	cheboratev density theorem

Hilbert's reciprocity	Artin's reciprocity
abc conjecture	Mason–Stothers_theorem
falting's theorem	function field falting's theorem+
chinese remainder	Lagrange's interpolation
Continued fraction	Padé table
Bézout's identity	Polynomial gcd++

	Riemann zeta	P-adic L-function
	Modular form	P-adic modular form
	Taniyama-Shimura conjecture	Iwasawa conjecture¿?
	Ideal class group+	Tate-Shafarevich group
	Chinese remainer theorem	Hasse principle
	Chowla-Selberg formula	Gross-Koblitz formula
	Multiplication theorem	Hasse-Davenport relation
	Gamma function	Gauss sump-adic Γ+
	Beta function	Jacobi sum
	Newton's method	Hensel's_lemma
	Complex analytic space	Rigid analytic space
	Hodge theory	P-adic Hodge theory
	Chebotarev's density theorem	Grothendieck-Katz p-curvature conjecture
	Uniformization	p-adic Uniformization
	derivative	p-derivative
	Borel subgroup	Iwahori subgroup

---

---

https://en.wikipedia.org/wiki/Siegel's_theorem_on_integral_points https://en.wikipedia.org/wiki/Mordell-Weil_theorem

---

similtonics

---

(Farkas's lemma-Fredholm alternative) (sdf matrix-sdf kernel) (Perron–Frobenius theoremKrein–Rutman theorem)

---

https://mathoverflow.net/questions/322590/analogy-between-metric-space-completion-and-algebraic-closure

---

connected set	connected graph+	Field
zeta		group zeta++

---

	g=0	g=1	g>1
homeomorphism	$\mathbb {S} ^{2}$	$\mathbb {T}$	$\#^{n}\mathbb {T}$
Universal cover	$\mathbb {S} ^{2}$	$\mathbb {R} ^{2}$	$\mathbb {H} ^{2}$
fundamental group	0	$\mathbb {Z} ^{2}$
automorphism	$PGL(2,\mathbf {C} )$		Hurwitz's bound
moduli	Example	j-invariat/cross ratio	3g-3
curvature	0	1	-1
Diophantine		Mordell-Weil theorem[7]+	Faltings's theorem

$\Delta \cong \mathbf {H} \subset \mathbf {C} \subset {\widehat {\mathbf {C} }}$(complex disk,plane,sphere) +

automorphisms	${\frac {az+b}{cz+d}}$	$ad-bc$
$\Delta$	${\frac {az+b}{{\overline {a}}z+{\overline {b}}}}$	$|a|^{2}-|b|^{2}=1$
$\mathbf {H}$	${\frac {az+b}{cz+d}}$	$ad-bc=1$
$\mathbf {C}$	$az+b$	$a\neq 0$
${\widehat {\mathbf {C} }}$	${\frac {az+b}{cz+d}}$	$ad-bc\neq 0$

+

---

Uniformization

---

conformal uniformization isothermal isothermal/conformal? Ricci flow elliptic uniformization +

---

Manifolds

https://en.wikipedia.org/wiki/Manifold_decomposition https://en.wikipedia.org/wiki/Category:3-manifolds https://en.wikipedia.org/wiki/Category:Structures_on_manifolds injectivity+1=surjectivity

Poincaré-Thurston

n>0:TO P=PL=DIFF
n>3:TOP=\=PL=DIFF
n>6:TOP=\=PL=\=DIFF

	True			False
	TOP	PL	DIFF	TOP	PL	DIFF
Manifold classification (MC)	1,2,3	1,2,3	1,2,3	n>3	n>3	n>3
Simple connected closed MC	1,2,3,5	1,2,3	1,2,3		n>3	n>3
Poincaré Conjecture	1,2,3,4,5,...	2,3,?,5,...	2,3,?,5,6			n>6

		TOP	PL	DIFF
n=2	Manifold classification (MC)	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=2	Simple connected closed MC	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=2	Poincaré conjecture	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=3	Manifold classification (MC)	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=3	Simple connected closed MC	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=3	Poincaré conjecture	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=4	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=4	Simple connected closed MC		False untriangulable & uncountable Casson & Donaldson	False untriangulable & uncountable Casson & Donaldson
n=4	Poincaré conjecture	True Cobordism Freedman
n=5	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=5	Simple connected closed MC	True Barden	False untriangulable Manolescu	False untriangulable Manolescu
n=5	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	True Cobordism Smale
n=6	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=6	Simple connected closed MC		False untriangulable Manolescu	False untriangulable Manolescu
n=6	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	True Cobordism Smale
n>=7	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n>=7	Simple connected closed MC		False untriangulable Manolescu	False untriangulable Manolescu
n>=7	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	False Exotic sphere Milnor

Template:Hidden end

---

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

---

---

---

Algebraic category: Donaldson–Thomas +

---

characterization: Gamma function+ Determinant exponential function

---

Analogies: Quine McCluskey algorithmBuchberger's algorithm

---

Cobordism

Thom Transversality theorem Thom space h-cobordism

Obstructions

Obstructions characteristic classes:

• homotopy=simple homotopy: Whitehead torsion+(homotopy groups equality=\=> homotopy equivalents)
• homeorphism=PL homeomorphism: Kirby-Siebenmann_class+
• homeomorphims=homotopic homeomorphism:Reidemeister torsion
• homeomosphism=homological homeomorphism?:Euler class
• vector bundle morphism=vector bundle isomorphism: Stiefel-Whitney_class
• complex vector bundle morphism=complex vector bundle isomorphism:Chern-class (unitary group)

+ Pontryagin_class (orthogonal group) ++ +

• Surgery obstruction+

• thom spectrum lack
• Todd classreciprocal characteristic class
• Segre class-inverse chern class
• Hasse invariant
• Manin obstruction
• Kronheimer-Mrowka basic class

---

Knots

https://en.wikipedia.org/wiki/Volume_conjecture https://en.wikipedia.org/wiki/Khovanov_homology#The_relation_to_link_(knot)_polynomials

---

---

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

---

Bott periodicity&Hurwitz's_theorem Bott_periodicity_theorem Hurwitz theorem Frobenius theorem

---

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

---

Hesisenberg group transform Special linear group transform +?

---

Special_unitary_group (spin group) + +2 +3 +4 +5

---

String group + +2 anomaly conformal invariants + +2

---

• soliton (localised in space)(fixed points of the flow?)
• breather (periodic soliton), oscilon (standing periodic soliton), kink(steady state periodic soliton)
• ADHM instanton
• Instanton(localised in time soliton)(1d:tunneling,2d:vortices,3d:higgs field-TP monopole, 4d: gauge bundle)
• Renormalon(Borel istanton)
• Cotangent localization

---

Helmholtz decomposition FLRW decomposition Ricci tensor decomposition Riemann tensor decomposition Electromagnetic tensor decomposition+

---

cauchy-pompieu decomposition poloidal decomposition clebsch decomposition

---

Sokhotski Plemelj theorem + Lippmann-Schwinger_equation

---

Covariance matrix Covariance process +

---

Var(x)=2Dt + + +

---

Cayley-Hamilton theorem Fredholm determinant +

isoperimetric&weyl stuff

Eigenvalues and eigenvectors of the second derivative

---

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

---

https://www.encyclopediaofmath.org/index.php/Blaschke-Santaló_inequality $T\cdot (KA)\geq L^{2}$+

---

Poincaré constant Cheeger constant Dirichlet eigenvalue$\lambda _{n+1}^{Neumann}<\lambda _{n}^{Dirichlet}$+

---

$N(\lambda )\leq {\frac {\lambda ab}{4\pi }}$+

---

$E(\lambda _{max}(A))\leq {\frac {1}{2}}E(\lambda _{max}(A-{\text{diag}}(u)))+{\frac {1}{2}}E(\lambda _{max}(A-{\text{diag}}(-u)))$++

---

Coarea formula Smooth coarea formula coarea+ coarea+

---

Gaussian isoperimetric inequality

variational quotients

https://en.wikipedia.org/wiki/Variational_inequality https://en.wikipedia.org/wiki/Complementarity_theory

---

Rayleigh quotient Variational method Fisher's linear discriminant Bauer Fike theorem

---

duality:
Min-max theorem Minimax theorem Max-min inequality Duality gap Lagrange multiplier

---

Geometric_phase Fubini-Study_metric Fisher_information_metric

---

Distance preserving Angle preserving Area preserving Volume preserving + Sympletic preserving + +

---

Knot group Link group Braid group

---

Knot theory Link theory Braid theory Ribbon theory +

---

Homology group Homotopy group Holonomy group

---

Cohomology group Cohomotopy group

---

Homological algebra Homotopical algebra

---

stable unitary group stable orthogonal group J-homomorphism

---

Inclusion-exclusion principle Isomorphism theorems

---

• atiyah-singer proof techniques: pseudodifferential operator, cobordism, k theory, heat operator

Manifold decomposition

• https://en.wikipedia.org/wiki/Borel_summation (best summation)
• https://en.wikipedia.org/wiki/Padé_approximant (best rational approximant)

• https://en.wikipedia.org/wiki/Category:Mathematics-related_lists
• https://en.wikipedia.org/wiki/List_of_cohomology_theories +
• https://en.wikipedia.org/wiki/Vanishing_theorem
• https://en.wikipedia.org/wiki/Fixed-point_theorem#List_of_fixed-point_theorems

+

• https://en.wikipedia.org/wiki/List_of_zeta_functions ++
• https://en.wikipedia.org/wiki/Trace_formula
• https://en.wikipedia.org/wiki/List_of_dualities
• https://en.wikipedia.org/wiki/Category:Duality_theories
• https://en.wikipedia.org/wiki/List_of_complex_and_algebraic_surfaces
• https://en.wikipedia.org/wiki/List_of_small_groups
• https://en.wikipedia.org/wiki/List_of_prime_knots (https://en.wikipedia.org/wiki/Knot_tabulation)
• https://en.wikipedia.org/wiki/List_of_mathematical_knots_and_links
• https://en.wikipedia.org/wiki/List_of_manifolds

---

• complexity classes lists: list complexity classes+complexity zoo

$\det(\exp(A))=\exp(\mathrm {tr} (A))$

$\ln(\det(A))=\mathrm {tr} (\ln(A))$

[https://en.wikipedia.org/wiki/Schrödinger_equation#Time_indep endent $\mathrm {\hat {H}} \Psi =E\Psi$]

---

Twelvefold way +

---

Plane partition number partition Multiplicative_partition (unordered factorization)

---

[8]

---

Numerical methods: Garlekin Homotopy analysis Finite difference Finite element Finite volume ≈

PHYSICSPHYSICSPHYSICSPHYSICSPHYSICS

$F=E+B\wedge dt$ + Maxwell's equations-Alternative formulations Mathematical descriptions of the electromagnetic field P-form electrodynamics Covariant formulation of classical electromagnetism

---

statistical physics

$-{\frac {F}{kT}}=\ln \operatorname {Tr} \exp {\big (}-{\tfrac {1}{kT}}{\hat {H}}{\big )}$
$f(E;\beta )=\beta {\frac {(\beta E)^{m-1}}{\Gamma (m)}}e^{-\beta E}\quad$ $P(N)={\frac {\langle N\rangle ^{N}\exp(-\langle N\rangle )}{N!}}={\frac {(\lambda t)^{N}\exp(-\lambda t)}{N!}}.$

---

+ + + + +

---

https://en.wikipedia.org/wiki/Bose_gas https://en.wikipedia.org/wiki/Gas_in_a_box https://en.wikipedia.org/wiki/Gas_in_a_harmonic_trap

---

physics stack

$\partial _{t}\psi =-\Gamma {\dfrac {\delta {\mathcal {F}}}{\delta \psi }}\quad$ ${\frac {\partial {\psi }}{\partial {t}}}=\nabla ^{2}{\dfrac {\delta {\mathcal {F}}}{\delta \psi }}.$ +

---

$U(T,0)={\cal {T}}(e^{-i\int _{0}^{T}H(t')dt'})=e^{-iTH_{\mathrm {e} ff}}$+

---

Conservation stuff

Class	Invariance	Conserved quantity
Proper orthochronous Lorentz symmetry	translation in time   (homogeneity)	energy
	translation in space   (homogeneity)	linear momentum
	rotation in space   (isotropy)	angular momentum
Discrete symmetry	P, coordinate inversion	spatial parity
	C, charge conjugation	charge parity
	T, time reversal	time parity
	CPT	product of parities
Internal symmetry (independent of spacetime coordinates)	U(1) gauge transformation	electric charge
	U(1) gauge transformation	lepton generation number
	U(1) gauge transformation	hypercharge
	U(1)Y gauge transformation	weak hypercharge
	U(2) [ U(1) × SU(2) ]	electroweak force
	SU(2) gauge transformation	isospin
	SU(2)L gauge transformation	weak isospin
	P × SU(2)	G-parity
	SU(3) "winding number"	baryon number
	SU(3) gauge transformation	quark color
	SU(3) (approximate)	quark flavor
	S(U(2) × U(3)) [ U(1) × SU(2) × SU(3) ]	Standard Model

+

Conservation Law	Respective Noether symmetry invariance		Number of dimensions
Conservation of mass-energy	Time invariance	Lorentz invariance symmetry	1	translation about time axis
Conservation of linear momentum	Translation symmetry		3	translation about x,y,z position
Conservation of angular momentum	Rotation invariance		3	rotation about x,y,z axes
CPT symmetry (combining charge conjugation, parity and time reversal)	Lorentz invariance		1 + 1 + 1	(charge inversion q → −q) + (position inversion r → −r) + (time inversion t → −t)
Conservation of electric charge	Gauge invariance		1⊗4	scalar field (1D) in 4D spacetime (x,y,z + time evolution)
Conservation of color charge	SU(3) Gauge invariance		3	r,g,b
Conservation of weak isospin	SU(2)L Gauge invariance		1	weak charge
Conservation of probability	Probability invariance		1 ⊗ 4	total probability always = 1 in whole x,y,z space, during time evolution

Three plane trigonometries

Trigonometry	Circular	Parabolic	Hyperbolic
Kleinian Geometry	Euclidean plane	Galilean plane	Minkowski plane
Symbol	E2	E0,1	E1,1
Quadratic form	positive definite	degenerate	non-degenerate but indefinite
Isometry group	E(2)	E(0,1)	E(1,1)
Isotropy group	SO(2)	SO(0,1)	SO(1,1)
type of isotropy	rotations	shears	boosts
Algebra over R	complex numbers	dual numbers	split-complex numbers
ε2	-1	0	1
Spacetime interpretation	none	Newtonian spacetime	Minkowski spacetime
slope	tan φ = m	tanp φ = u	tanh φ = v
"cosine"	cos φ = (1+m2)−1/2	cosp φ = 1	cosh φ = (1-v2)−1/2
"sine"	sin φ = m (1+m2)−1/2	sinp φ = u	sinh φ = v (1-v2)−1/2
"secant"	sec φ = (1+m2)1/2	secp φ = 1	sech φ = (1-v2)1/2
"cosecant"	csc φ = m−1 (1+m2)1/2	cscp φ = u−1	csch φ = v−1 (1-v2)1/2

++

---

https://en.wikipedia.org/wiki/Curvature_form#Bianchi_identities

---

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

---

NC

https://en.wikipedia.org/wiki/Noncommutative_quantum_field_theory https://en.wikipedia.org/wiki/IR/UV_mixing https://en.wikipedia.org/wiki/Kramers-Wannier_duality

Ising stuff

Kramers-Wannier duality IR/UV mixing +

homotophysics

$\pi _{3}\left({\frac {SU(N)_{L}\times SU(N)_{R}}{SU(N)_{\text{diag}}}}\cong SU(N)\right)=\mathbf {Z}$ + $\pi _{2}\left({\frac {SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbf {Z} _{3}}}\right)=\mathbf {Z}$ + $\pi _{4}(SU(2))=\mathbf {Z} /2\mathbf {Z}$ +

Black hole metrics

	Non-rotating (J = 0)	Rotating (J ≠ 0)
Uncharged (Q = 0)	Schwarzschild	Kerr
Charged (Q ≠ 0)	Reissner–Nordström	Kerr–Newman

Alternative coordinates

Coordinates	Line element	Notes	Features
Eddington–Finkelstein coordinates (ingoing)	$-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dv^{2}+2\,dv\,dr+r^{2}\,g_{\Omega }$		regular at future horizon --past horizon is at v=- infinity
Eddington–Finkelstein coordinates (outgoing)	$-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,du^{2}-2\,du\,dr+r^{2}g_{\Omega }$		regular at past horizon extends across past horizon. Future horizon at u = infinty
Gullstrand–Painlevé coordinates	$-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\,dT^{2}\pm 2{\sqrt {\frac {r_{\mathrm {s} }}{r}}}\,dT\,dr+dr^{2}+r^{2}\,g_{\Omega }$		regular at (+ future/-past) horizon
Isotropic coordinates	$-{\frac {\left(1-{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}{\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{2}}}\,{dt}^{2}+\left(1+{\frac {r_{\mathrm {s} }}{4R}}\right)^{4}\,\left(dx^{2}+dy^{2}+dz^{2}\right)$	$R={\sqrt {x^{2}+y^{2}+z^{2}}}$ Valid only outside the event horizon: $R>r_{s}/4$	isotropic lightcones on constant time slices
Kruskal–Szekeres coordinates	$-{\frac {4r_{\mathrm {s} }^{3}}{r}}e^{-{\frac {r}{r_{\mathrm {s} }}}}\,\left(dT^{2}-dR^{2}\right)+r^{2}\,g_{\Omega }$	$T^{2}-R^{2}=\left(1-{\frac {r}{r_{\mathrm {s} }}}\right)e^{\frac {r}{r_{\mathrm {s} }}}$	regular at horizon Maximally extends to full spacetime
Lemaître coordinates	$-dT^{2}+{\frac {r_{\mathrm {s} }}{r}}\,dR^{2}+r^{2}\,g_{\Omega }$	$r=\left({\tfrac {3}{2}}(R\pm T)\right)^{\frac {2}{3}}r_{\mathrm {s} }^{\frac {1}{3}}$	regular at future/past horizon

+ + + + + + +

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Precession

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Thomas precession	Gravitational Thomas precession
Larmor precession++?	de Sitter precession (Lense-Thirring precession)
Total angular momentum quantum operator	Total angular momentum tensor
Einstein–de Haas effect Barnett effect	Landau–Lifshitz–Gilbert equation

Precession +

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Bohr-Sommerfeld EBK-Bohr-sommerfeld cyclotron Bohr-Sommerfeld

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Madelung equations Circulation

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Aharonov-Bohm effect Landau quantization

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Wilson loop t'hooft loop

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https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)

---

https://physics.stackexchange.com/questions/295714/whats-the-relation-between-path-integral-and-dyson-series http://bolvan.ph.utexas.edu/~vadim/Classes/2011f/dyson.pdf

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$S_{q}=S_{q}^{FD}+S_{q}^{BE}$ + $S_{\text{B}}=S_{s}+k\log \omega .$ + $\Delta H\equiv \Delta H_{\text{L}}+\Delta H_{\text{T}}$ + ${\mathcal {H}}={\mathcal {H}}_{\rm {Coulomb}}+{\mathcal {H}}_{\mathrm {kinetic} }+{\mathcal {H}}_{\mathrm {SO} }+H_{\mathrm {Darwin} }\!$ + + ${\hat {H}}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}+{\hat {H}}_{\text{int}}$ + $H=D_{1}+D_{2}+D_{3}$ + $E_{\mathrm {total} }=E_{\mathrm {electronic} }+E_{\mathrm {vibrational} }+E_{\mathrm {rotational} }+E_{\mathrm {nuclear\,spin} }$ + $\epsilon =\epsilon _{elec}+\epsilon _{vib}+\epsilon _{rot}+\epsilon _{trans}$ + $\sigma =\sigma _{\text{a}}+\sigma _{\text{s}}+\sigma _{\text{l}}.$ + ${\frac {1}{\tau }}={\frac {1}{\tau _{\rm {impurities}}}}+{\frac {1}{\tau _{\rm {lattice}}}}+{\frac {1}{\tau _{\rm {defects}}}}+\cdots$. + ${\frac {1}{\lambda _{\mathrm {MFP} }}}={\frac {1}{\lambda _{\mathrm {el-el} }}}+{\frac {1}{\lambda _{\mathrm {ap} }}}+{\frac {1}{\lambda _{\mathrm {op,ems} }}}+{\frac {1}{\lambda _{\mathrm {op,abs} }}}+{\frac {1}{\lambda _{\mathrm {impurity} }}}+{\frac {1}{\lambda _{\mathrm {defect} }}}+{\frac {1}{\lambda _{\mathrm {boundary} }}}$ + Failed to parse (syntax error): {\displaystyle ''ω''_total {{=}} ''ω''_{General Relativity} + ''ω''_quadrupole + ''ω''_tide + ''ω''_{perturbations}} +

---

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

---

Dilaton Perelman's renormalization Perleman's entropy Perelman's fluctuation Quantropy Jacobson's entropic field Verlinde's entropic force [9]

---

Jacobson	$T={\frac {\kappa }{2\pi }}$	$S={\frac {k_{\mathrm {B} }A}{4l_{P}^{2}}}$	$\delta Q=TdS$	$R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$
Verlinde	$T={\frac {\hbar GM}{2\pi ck_{\mathrm {B} }r^{2}}}$	$S={\frac {2\pi k_{\mathrm {B} }mc}{\hbar }}$	$F=T\nabla S$	$F=G\,{\frac {Mm}{r^{2}}}$

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velocity limit: light velocity entropy limit: bekenstein bound

---

Unruh radiation Hawking radiation +

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+ Locality breaker: EPR paradox Unitarity breaker: BHI paradox causality breaker: relativity

---

Newton	----	HJ	Koopman–von Neumann
Ehrenfest theorem	Schrodinger	Bhom	Moyal
GR	ADM	EHJ	chern-simmonns?

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Spacetime symmetries

---

Sagnac effect

---

YB equation BMW algebra +

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ΛCDM common observations: SNIa CMB H(z) BAO LSS
Planck spacecraft + standard candle standard ruler

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Hamilton Jacobi_equation Circulation Biot-Savart

---

Kutta Joukowski theorem Magnus effect Vorticity equation

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Kelvin's circulation theorem tao's post Euler fluid equations Hamiltonian fluid mechanics Madelung equations

---

---

P: Conserved quantities

Laplace–Runge–Lenz vector hidrodinamical helicity Enstrophy

---

P: Electronic transport mechanisms

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Electronic transport mechanisms

Transport mechanism	Effect of electric field	Functional form	Differential form
Fowler-Nordheim tunneling (field emission)Template:Ref		$I=A_{\text{eff}}{\frac {e^{3}m}{8\pi hm\phi _{\text{B}}}}E^{2}\exp \left(-{\frac {8\pi {\sqrt {2m}}}{3he}}{\frac {\phi _{\text{B}}^{3/2}}{E}}\right)$	${\frac {\mathrm {d} \ln I}{\mathrm {d} V}}={\frac {1}{V}}+{\frac {4\pi }{3he}}{\sqrt {2m}}{\frac {\phi _{\text{B}}^{3/2}}{V^{2}}}$
Thermionic emissionTemplate:Ref	Lowers barrier height	$I=A_{\text{eff}}A^{\star }T^{2}\exp \left[-{\frac {e}{k_{\text{B}}T}}\left(\phi _{\text{B}}-{\sqrt {\frac {eE}{4\pi \epsilon _{0}\epsilon _{r}}}}\right)\right]$	${\frac {\mathrm {d} \ln I}{\mathrm {d} V}}={\frac {e}{4k_{\text{B}}T}}\left({\frac {e}{\pi \epsilon _{0}\epsilon _{r}}}\right)^{\frac {1}{2}}V^{-{\frac {1}{2}}}$
Arrhenius equationTemplate:Ref		$I=e\mu nE\exp \left(-{\frac {E_{a}}{k_{\text{B}}T}}\right)$	${\frac {\mathrm {d} \ln I}{\mathrm {d} E}}={\frac {1}{E}}$
Poole–Frenkel hopping	Assists thermal ionization of trapped charges	$I=e\mu nE\exp \left[-{\frac {e}{k_{\text{B}}T}}\left(\phi _{\text{B}}-{\sqrt {\frac {eE}{4\pi \epsilon _{0}\epsilon _{r}}}}\right)\right]$	${\frac {\mathrm {d} \ln I}{\mathrm {d} E}}={\frac {1}{E}}+{\frac {e}{4k_{\text{B}}T}}\left({\frac {e}{\pi \epsilon _{0}\epsilon _{r}}}\right)^{\frac {1}{2}}E^{-{\frac {1}{2}}}$
Thermally-assisted tunnelingTemplate:Ref		$I={\frac {V}{2\pi }}\left({\frac {k_{\text{B}}Tt}{2\pi }}\right)^{\frac {1}{2}}\exp \left(-{\frac {\phi }{k_{\text{B}}T}}+{\frac {V^{2}\Theta }{24\left(k_{\text{B}}T\right)^{3}}}\right)$	${\frac {\mathrm {d} \ln I}{\mathrm {d} V}}={\frac {1}{V}}+{\frac {V\Theta }{12\left(k_{\text{B}}T\right)^{3}}}$

+

---

+Hot-carrier injection Ballistic conduction

---

Field electron emission

---

P: Optics

---

https://en.wikipedia.org/wiki/Brewster's_angle

---

P: diffraction, refraction, reflection, interference stuff

---

https://en.wikipedia.org/wiki/Kirchhoff's_diffraction_formula https://en.wikipedia.org/wiki/Fresnel_diffraction + https://en.wikipedia.org/wiki/Fraunhofer_diffraction

---

https://en.wikipedia.org/wiki/N-slit_interferometric_equation

---

$\mathbf {\hat {d}} _{\mathrm {s} }=\mathbf {R} \;\mathbf {\hat {d}} _{\mathrm {i} }=(\mathbf {I} -2\mathbf {\hat {d}} _{\mathrm {n} }\mathbf {\hat {d}} _{\mathrm {n} }^{\mathrm {T} })\;\mathbf {\hat {d}} _{\mathrm {i} }$ +

---

https://en.wikipedia.org/wiki/Dynamical_theory_of_diffraction https://en.wikipedia.org/wiki/Diffraction_formalism

---

P: scattering & cross section

https://en.wikipedia.org/wiki/Scattering https://en.wikipedia.org/wiki/Scattering_theory https://en.wikipedia.org/wiki/Category:Scattering +

---

Elastic:
https://en.wikipedia.org/wiki/Rayleigh_scattering https://en.wikipedia.org/wiki/Mie_scattering https://en.wikipedia.org/wiki/Compton_scattering (https://en.wikipedia.org/wiki/Thomson_scattering) https://en.wikipedia.org/wiki/Rutherford_scattering
Inelastic:
https://en.wikipedia.org/wiki/Raman_scattering https://en.wikipedia.org/wiki/Deep_inelastic_scattering
+:
https://en.wikipedia.org/wiki/Two-photon_physics https://en.wikipedia.org/wiki/Delbrück_scattering https://en.wikipedia.org/wiki/Quantum_mechanical_scattering_of_photon_and_nucleus

---

https://en.wikipedia.org/wiki/Cross_section_(physics) https://en.wikipedia.org/wiki/Scattering_cross-section https://en.wikipedia.org/wiki/Nuclear_cross_section https://en.wikipedia.org/wiki/Neutron_cross_section http://hep.physics.wayne.edu/~harr/courses/5210/w15/lecture29.htm

---

https://en.wikipedia.org/wiki/Moiré_pattern https://en.wikipedia.org/wiki/Scattering_amplitude https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem https://en.wikipedia.org/wiki/Mean_free_path https://en.wikipedia.org/wiki/Scattering_length

---

Electron scattering: +
https://en.wikipedia.org/wiki/Bhabha_scattering https://en.wikipedia.org/wiki/Møller_scattering https://en.wikipedia.org/wiki/Compton_scattering https://en.wikipedia.org/wiki/Mott_scattering https://en.wikipedia.org/wiki/Bremsstrahlung https://en.wikipedia.org/wiki/Deep_inelastic_scattering https://en.wikipedia.org/wiki/Synchrotron_emission

---

Light scattering: +
https://en.wikipedia.org/wiki/Rayleigh_scattering https://en.wikipedia.org/wiki/Geometric_optics https://en.wikipedia.org/wiki/Mie_scattering https://en.wikipedia.org/wiki/Umklapp_scattering

---

https://en.wikipedia.org/wiki/Scattering_parameters https://en.wikipedia.org/wiki/Forward_scattering_alignment https://en.wikipedia.org/wiki/Back_scattering_alignment

---

https://en.wikipedia.org/wiki/Lippmann–Schwinger_equation

---

${\frac {I_{0}}{I}}={\frac {\,d\Omega \,dA\cos(\theta )}{d\Omega _{0}\,\,dA_{0}\cos(\theta )}}={\frac {\,d\Omega \,dA}{d\Omega _{0}\,dA_{0}}},\,$ $F_{tot}=\int \limits _{0}^{2\pi }\,\int \limits _{0}^{\pi /2}\cos(\theta )I_{max}\,\sin(\theta )\,\operatorname {d} \theta \,\operatorname {d} \phi ,\,$ $\Phi _{\mathrm {e} }=\oint _{\Sigma }\mathbf {S} \cdot \mathbf {\hat {n}} \,\mathrm {d} A=\oint _{\Sigma }|\mathbf {S} |\cos \alpha \,\mathrm {d} A,\,$ $L_{\mathrm {e} ,\Omega }={\frac {\partial ^{2}\Phi _{\mathrm {e} }}{\partial \Omega \,\partial A\cos \theta }},$ $\mathbf {F} (\mathbf {x} ,t;\nu )=\oint _{\Omega }\ I(\mathbf {x} ,t;\mathbf {\hat {n}} ,\nu )\,\mathbf {\hat {n}} \,d\omega (\mathbf {\hat {n}} )$ + + + +

---

P: Crystallography stuff

---

Crystallographic periodic table crystal structure Crystall system some 3D point groups character tables Peierls stress

---

Bloch's theorem $H(r+R)=T_{R}H(R)T_{R}^{-1}$ Floquet theory

---

Crystal_momentum Umklapp_scattering

---

$\mathbf {M} _{\rm {orb}}={\frac {1}{2V}}\int _{V}d^{3}\mathbf {r} \,\mathbf {r} \times \mathbf {J} (\mathbf {r} ),\quad$ $\mathbf {M} _{\rm {orb}}={\frac {-e}{2m_{e}}}\sum _{n}\int _{\rm {BZ}}{\frac {d^{3}k}{(2\pi )^{3}}}\,\langle \psi _{n\mathbf {k} }\vert \mathbf {r} \times \mathbf {p} \vert \psi _{n\mathbf {k} }\rangle \,,$ +

---

Direct space	Bravais lattice	Wigner-Seitz_cell	$\mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}$
Reciprocal space	Recriprocal lattice	Brillouin zone	$\mathbf {G} =k_{1}\mathbf {b} _{1}+k_{2}\mathbf {b} _{2}+k_{3}\mathbf {b} _{3}$
			$\left[\mathbf {b} _{1}\mathbf {b} _{2}\mathbf {b} _{3}\right]^{\mathsf {T}}=2\pi \left[\mathbf {a} _{1}\mathbf {a} _{2}\mathbf {a} _{3}\right]^{-1}.$ $\mathbf {b} _{i}\cdot \mathbf {a} _{j}=2\pi \delta _{ij}$ $\mathbf {G} \cdot \mathbf {R} =2\pi (k_{1}n_{1}+k_{2}n_{2}+k_{3}n_{3})$ $e^{i\mathbf {G} \cdot \mathbf {R} }=1$

+ +

---

$\mathbf {\Delta k} \cdot \mathbf {x} =\mathbf {\Delta k} \cdot (p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} )=p\,2\pi h+q\,2\pi k+r\,2\pi l=2\pi (hp+kq+lr)=2\pi n,$ + $2d\sin \theta =n\lambda$ +

---

https://en.wikipedia.org/wiki/List_of_quasiparticles https://en.wikipedia.org/wiki/List_of_particles

---

Quark lepton_complementarityEigthfold way

---

mass oscilation	charge oscillation	spin oscilations
PhononRotonMaxon	polarons-exciton	magnons

Phonon-polaron-magnon spinon-orbiton-chargon https://en.wikipedia.org/wiki/Bose-Einstein_condensation_of_quasiparticles

---

Molecular partition function + + +

---

volumetric entropy surface entropy gas entropy + + +

---

+ thermodynamics +

---

Deconfinement

---

Maxwell-boltzman statistics	Bose-Einstein statistics	Fermi-Dirac statistics	anyon statistics	braid statistics
distinguishable	Indistinguishable	Indistinguishable	Indistinguishable	Indistinguishable
Pauli	degenerate	Pauli
classical gas	boson	fermion	anyon	plekton

https://en.wikipedia.org/wiki/Scale-free_ideal_gas

---

• electromagnetism
• relativistic charge distribution
• relativistic charge point
• +

---

wave: string, springs, bars

---

• https://en.wikipedia.org/wiki/Chapman-Enskog_theory(https://en.wikipedia.org/wiki/Navier-Stokes_equations https://en.wikipedia.org/wiki/Boltzmann_equation)

---

	Non-rotating (J = 0)	Rotating (J ≠ 0)
Uncharged (Q = 0)	Schwarzschild	Kerr
Charged (Q ≠ 0)	Reissner–Nordström	Kerr–Newman

---

Minkowski	$\Lambda =0$	$ISO(3,1)$
De Sitter	$\Lambda >0$	$SO(4,1)$
Anti De sitter	$\Lambda <0$	$SO(3,2)=SO(3,1)xSO(3,1)$?

++

---

Group contractionQuantization commutes with reduction

---

D'Alembert->KdV Hierarchy/Dirac operator->ANKNS hierarchyBBGKY hierarchy

---

Hamiltonian fluid mechanics Madelung equation De_Broglie–Bohm_theory Convection diffusion equation Langevin dynamics

---

Langevin motion + +

---

Electromagnetism Gravitoelectromagnetism + Kaluza-klein theory Yang-Mills theory

---

T*C->Poison bracket
T*M_g->ADM bracket
+

---

• https://en.wikipedia.org/wiki/Supersymmetry_nonrenormalization_theorems
• https://en.wikipedia.org/wiki/Loop_representation_in_gauge_theories_and_quantum_gravity

---

Black hole: Black hole radiation dark matter

---

Cosmic Backgrounds:
Relics: Photon(CMB) Neutrino Gravitation
DEBRA: Infrared X-ray Extragalactic light Radio Gamma-ray CMB

---

Non-linear

Primakoff_effectSchwinger limitEuler-Heisenberg Lagrangian

angle mechanism

Angles: Weinberg Peccei–Quinn GIM CKM PMKS + vacuum angle

---

Planck temperature Hagedorn temperature +

---

Noether's Theorem extensions Wald entropy formula++

---

stringification=categorization

---

---

Types of radioactive decay Template:Anchor

Mode of decay	Daughter nucleus
Decays with emission of nucleons:
Alpha decay	(A − 4, Z − 2)
Proton emission	(A − 1, Z − 1)
Neutron emission	(A − 1, Z)
Double proton emission	(A − 2, Z − 2)
Spontaneous fission	—
Cluster decay	(A − A1, Z − Z1) + (A1, Z1)
Different modes of beta decay:
β− decay	(A, Z + 1)
Positron emission (β+ decay)	(A, Z − 1)
Electron capture	(A, Z − 1)
Bound state beta decay	(A, Z + 1)
Double beta decay	(A, Z + 2)
Double electron capture	(A, Z − 2)
Electron capture with positron emission	(A, Z − 2)
Double positron decay	(A, Z − 2)
Transitions between states of the same nucleus:
Isomeric transition	(A, Z)
Internal conversion	(A, Z)

Nuclide radioactive decay map
Nucleosynthesis periodic table
[10]
Template:Nuclear physics

coding

atributes(class/intance), method (function atribute), paramenter, argument
"{}, you'll always be the {}th planet to me.".format(planet, position)="Pluto, you'll always be the 9th planet to me." +
type(),dir(),help() +
https://pyformat.info/ https://docs.python.org/3/

ctf

ctf0:ctrl+I->Network->crtl+f5->background.png