blablabla
Contents
1 start-93(11/10/2020)
2 polynomials
3 fourier vs contour
4 terms
5 weird
6 snippets
7 stack stuff
8 stack stuff 2
9 stuff
10 combinatorics
11 +++++++++++
12 Universality stuff
13 Correspondences&dictionaries
14 Generalizations
15 classification miscelanea
16 Invariant miscelanea
17 Enumerative invariants
18 duality miscelanea
19 algebraic geo stuff
20 category theory
21 discrete stuff
22 stackexchange
23 generating functions
24 diferential representations
25 Sequences
26 congmatics
27 Diagramatics
28 matrices stuff
29 Gaps
30 spectral stuff
31 linear algebra
32 stuff with det,tr
33
χ
{\displaystyle \chi}
34 "Index theory"
35 Algebraic Geometry
36 Riemann-Roch Stuff
37 Hodge stuff
38 Homotopy stuff
39 algebraic topology
40 geometric algebra
41 sympletic geometry
42 Differential geo
43 Degenerancy theory
44 Representation stuff
45 conmutative tuff
46 group stuff
47 mobius table
48 mobius+
49 Lambert
50 elliptic stuff
51 arithmetic
52 primes stuff
53 Moduli stuff
54 height stuff
55 class number
56 elliptic and quadratics
57 tuple primes
58 hypergeometric stuff
59 eta stuff
60 heat table
61 convergence stuff
62 LLN,LIL,CLT
63 PNT,RH,EK
64 numbers, fields, curves, p-adic
65 similtonics
66 Uniformization
67 Manifolds
68 Poincaré-Thurston
69 Cobordism
70 Obstructions
71 Knots
72 isoperimetric&weyl stuff
73 variational quotients
74 PHYSICSPHYSICSPHYSICSPHYSICSPHYSICS
75 statistical physics
76 physics stack
77 Conservation stuff
78 NC
79 Ising stuff
80 homotophysics
81 Black hole metrics
82 Precession
83 P: Conserved quantities
84 P: Electronic transport mechanisms
85 P: Optics
86 P: diffraction, refraction, reflection, interference stuff
87 P: scattering & cross section
88 P: Crystallography stuff
89 Non-linear
90 angle mechanism
91 Language
92 coding
93 ctf
start-93(11/10/2020)
polynomials
Chebotarev theorem on roots of_unity
generating question 1 gq 2 gq3 gq4
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
+
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
https://en.wikipedia.org/wiki/Category:Mathematics_navigational_boxes
stack stuff
∑
k
=
1
n
k
m
=
∑
b
=
1
m
+
1
(
n
b
)
∑
i
=
0
b
−
1
(
−
1
)
i
(
b
−
i
)
m
(
b
−
1
i
)
{\displaystyle \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}
+
∑
n
=
1
∞
ζ
(
2
n
)
x
2
n
=
−
π
x
2
cot
(
π
x
)
{\displaystyle \sum_{n=1}^\infty \zeta(2n)x^{2n} = -\frac{\pi x}{2}\cot(\pi x)}
+
toric=>rational+
∑
t
∈
F
p
n
ζ
T
r
(
t
)
=
0
{\displaystyle \sum_{t \in \mathbb{F}_{p^n}} \zeta ^{Tr (t)} = 0}
+
stack stuff 2
d
3
≠
a
3
+
b
3
+
c
3
⇔
∗
d
3
≡
4
,
5
(
mod
9
)
⇒
d
3
≡
1
,
2
(
mod
3
)
⇔
(
a
+
b
+
c
)
≡
1
,
2
(
mod
3
)
{\displaystyle d^3\neq a^3+b^3+c^3\Leftrightarrow_{*} d^3\equiv 4,5\pmod9 \Rightarrow d^3\equiv 1,2\pmod3\Leftrightarrow (a+b+c)\equiv 1,2\pmod3}
[1]
stuff
g
n
−
1
(
x
)
=
sup
i
≤
n
{
f
i
−
1
(
x
)
}
=
⋃
i
=
1
n
f
i
−
1
(
(
a
,
∞
)
)
{\displaystyle g_n^{-1}(x) = \sup_{i \leq n} \{f_i^{-1}(x)\}=\bigcup_{i=1}^n f_i^{-1}((a,\infty))}
+
k
∣
n
⇔
p
k
−
1
∣
p
n
−
1
⇔
x
p
k
−
x
∣
x
p
n
−
x
.
{\displaystyle 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
{\displaystyle \gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1}
+
gcd
(
m
,
n
)
=
∏
p
i
min
(
m
i
,
n
i
)
{\displaystyle \gcd(m,n)=\prod p_i^{\min(m_i,n_i)}}
x
2
≡
a
mod
p
⇔
a
p
−
1
2
≡
1
mod
p
{\displaystyle x^2 \equiv a \mbox{ } \mbox{mod} \mbox{ } p\Leftrightarrow a^\frac{p-1}{2} \equiv 1\mbox{ } \mbox{mod} \mbox{ } p}
+
(
x
+
y
i
)
p
≡
x
p
+
y
p
i
p
≡
x
+
(
−
1
)
p
−
1
2
y
i
(
mod
p
)
{\displaystyle (x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p}}
+
|
a
→
b
→
c
→
|
=
a
→
⋅
(
b
→
×
c
→
)
{\displaystyle \left|\vec a\ \vec b\ \vec c\right|=\vec a \cdot (\vec b \times \vec c)}
+
d
Ω
=
4
π
(
d
Σ
A
)
(
r
^
⋅
n
^
)
=
4
π
(
d
Σ
4
π
r
2
)
(
r
^
⋅
n
^
)
{\displaystyle 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})}
+
d
q
d
t
=
∬
S
j
⋅
n
^
d
A
=
a
r
g
m
a
x
n
^
n
^
p
d
q
d
t
(
A
,
p
,
n
^
)
{\displaystyle \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}})}
+
c
+
d
i
c
−
d
i
=
c
2
−
d
2
c
2
+
d
2
+
2
c
d
c
2
+
d
2
i
{\displaystyle \frac{c+di}{c-di}=\frac{c^2-d^2}{c^2+d^2} + \frac{2cd}{c^2+d^2} i}
[2]
https://en.wikipedia.org/wiki/Zsigmondy's_theorem
Cotangent Bundles
↔
Pull-backs
↔
Differentials
{\displaystyle \text{Cotangent Bundles}\leftrightarrow \text{Pull-backs}\leftrightarrow \text{Differentials}}
Tangent Bundles
↔
Push-forward
↔
Tangent Vectors
{\displaystyle \text{Tangent Bundles}\leftrightarrow \text{Push-forward}\leftrightarrow \text{Tangent Vectors}}
+
Hausdorff moment problem
Stieljes moment problem
Hamburger moment problem
https://en.wikipedia.org/wiki/Highly_composite_number
https://en.wikipedia.org/wiki/Van_Eck's_sequence
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
https://en.wikipedia.org/wiki/Frobenius_reciprocity
https://en.wikipedia.org/wiki/Cyclotomic_polynomial
https://proofwiki.org/wiki/Reciprocals_of_Odd_Numbers_adding_to_1
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
https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities
https://en.wikipedia.org/wiki/Generating_function
combinatorics
(
n
k
)
=
1
2
π
i
∮
|
z
|
=
1
(
1
+
z
)
n
z
k
+
1
d
z
{\displaystyle \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
e
∑
k
=
0
∞
k
n
k
!
.
{\displaystyle B_n = {1 \over e}\sum_{k=0}^\infty \frac{k^n}{k!}.}
+
{
j
k
}
=
1
k
!
∑
i
=
0
k
(
k
i
)
(
−
1
)
i
(
k
−
i
)
j
{\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
)
=
n
!
2
π
i
∮
γ
f
(
z
)
(
z
−
a
)
n
+
1
d
z
.
{\displaystyle f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.}
+
f
−
(
n
+
1
)
(
x
)
=
1
n
!
∫
a
x
(
x
−
t
)
n
f
(
t
)
d
t
{\displaystyle f^{-(n+1)}(x)= \frac{1}{n!} \int_a^x \left(x-t\right)^n f(t)\,\mathrm{d}t}
+
∮
γ
(
z
−
c
)
n
−
k
−
1
d
z
=
2
π
i
δ
n
k
{\displaystyle \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
https://en.wikipedia.org/wiki/Foias_constant
Correspondences&dictionaries
https://ncatlab.org/nlab/search?query=correspondence +
Schreiber's correspondences+
Correspondences:
algebraic sets & Ideals
Field subextension & Galois Subgroups
Galois group & Fundemantal group
+
+
+
algebra/geometry & galois/fundamental group
Modular forms & Elliptic curves
Automorphic forms & Algebraic curves
Elliptic modular forms & Group representations
Geometric langlands
Kobayashi–Hitchin correspondence
Simpson correspondence
Riemann-Hilbert correspondence
Robinson-Schensted correspondence
Shimura correspondence -Theta correspondence
Haussdorf-C* algebra duality
+
+
+
Lawrence theory
Morita's equivalence
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 +
votja's conjecture +
Baez-Stay dictionary
https://en.wikipedia.org/wiki/Category:Duality_theories
https://en.wikipedia.org/wiki/Lie_group-Lie_algebra_correspondence#Proof_of_the_homomorphisms_theorem
Generalizations
(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 theorem Malgrange preparation theorem )
classification miscelanea
https://en.wikipedia.org/wiki/Uniqueness_theorem
+
+
+
Invariant miscelanea
(Invariant theory -Geometric invariant theory )
+
+
https://en.wikipedia.org/wiki/Invariant_differential_operator
https://en.wikipedia.org/wiki/Differential_invariant
https://en.wikipedia.org/wiki/Differential_operator
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
+
+
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
μ
(
E
)
=
d
e
g
E
r
k
E
{\displaystyle \mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}\quad}
μ
(
E
)
=
d
i
m
H
0
(
X
,
E
)
−
d
i
m
H
1
(
X
,
E
)
r
a
n
k
E
+
g
X
−
1
{\displaystyle \mu(E)=\frac{dim H^0(X,E)-dim H^1(X,E)}{rank E}+g_X-1}
+
+
+
category theory
five lemma six operations
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
+
(
Z
,
ker
(
f
,
g
)
)
≃
ker
(
Hom
(
Z
,
X
)
⇉
Hom
(
Z
,
Y
)
)
{\displaystyle (Z, \ker(f, g)) ≃ \ker(\text{Hom} (Z, X) ⇉ \text{Hom} (Z, Y ))}
+
discrete stuff
discrete taylor series
discrete taylor series table
discrete integration by parts
discrete laplacian
discrete spectral theory
stackexchange
The Selberg trace formula is making
P
S
L
(
2
,
R
)
{\displaystyle PSL(2,\Bbb{R})}
act on
C
∞
(
Γ
∖
H
)
{\displaystyle C^\infty(\Gamma \setminus \Bbb{H})}
, the Frobenius formula is making G act on
C
[
G
/
H
]
{\displaystyle \Bbb{C}[G/H]}
+
generating functions
Generating functions +
∑
n
=
0
∞
s
(
n
)
n
!
x
n
=
(
1
−
x
2
)
−
1
4
exp
(
x
2
4
)
.
{\displaystyle \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
∑
n
=
0
∞
∑
k
=
0
n
B
(
k
,
n
)
=
∑
n
=
0
∞
∑
k
=
0
∞
B
(
k
,
n
+
k
)
{\displaystyle \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
e
D
2
f
(
x
)
=
∑
k
=
0
∞
D
2
k
k
!
f
(
x
)
.
{\displaystyle e^{D^2} f(x) = \sum_{k=0}^\infty \frac{D^{2k}}{k!}f(x).}
e
D
−
1
D
f
(
x
)
=
∑
n
=
0
∞
D
n
(
n
+
1
)
!
f
(
x
)
{\displaystyle {e^D - 1 \over D}f(x)= \sum_{n=0}^\infty {D^n \over (n+1)!}f(x)}
e
D
2
f
(
x
)
=
1
4
π
∫
−
∞
∞
f
(
x
−
y
)
e
−
y
2
/
4
d
y
{\displaystyle e^{D^2}f(x)=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y) e^{-y^2/4}\;dy}
e
D
−
1
D
f
(
x
)
=
∫
x
x
+
1
f
(
u
)
d
u
{\displaystyle {e^D - 1 \over D}f(x) = \int_x^{x+1} f(u)\,du}
e
−
D
2
∑
n
=
0
∞
a
n
x
n
=
∑
n
=
0
∞
a
n
H
n
(
x
/
2
)
{\displaystyle e^{-D^2}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n H_n(x/2)}
D
e
D
−
1
∑
n
=
0
∞
a
n
x
n
=
∑
n
=
0
∞
a
n
B
n
(
x
)
{\displaystyle {D \over e^D -1}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n B_n(x)}
+
+
+
D
e
D
e
D
−
1
=
∑
n
=
0
∞
B
n
n
!
D
n
=
∑
n
=
0
∞
(
−
1
)
n
−
1
ζ
(
1
−
n
)
(
n
−
1
)
!
D
n
=
−
1
+
D
2
−
D
2
12
+
.
.
.
{\displaystyle \frac{D e^D}{e^D-1}=\sum_{n=0}^\infty \frac{B_n}{n!} D^n=\sum_{n=0}^{\infty} (-1)^{n-1} \frac{\zeta(1-n)}{(n-1)!}D^{n}=-1+\frac{D}{2}-\frac{D^2}{12}+...}
+ +
+
e
t
D
f
(
x
)
=
f
(
x
+
t
)
{\displaystyle e^{t D}f(x)= f(x+t)}
+ +
+
(
X
+
D
)
n
=
∑
j
=
0
n
(
n
j
)
H
n
−
j
(
X
)
D
j
{\displaystyle (X + D)^n = \sum_{j = 0}^{n }{n\choose j} H_{n-j}(X)D^j}
+
(
1
−
D
)
2
X
t
=
X
t
−
2
X
t
−
1
+
X
t
−
2
{\displaystyle (1-D)^2X_t = X_t -2X_{t-1} + X_{t-2}\quad}
(
1
−
B
)
d
=
∑
k
=
0
∞
(
d
k
)
(
−
B
)
k
{\displaystyle (1 - B)^d= \sum_{k=0}^{\infty} \; {d \choose k} \; (-B)^k}
+
Sequences
https://en.wikipedia.org/wiki/Category:Spectral_sequences
Surgery exact sequence
Serre spectral sequence
Mapping_class_group
https://en.wikipedia.org/wiki/Poisson_summation_formula#Derivation
congmatics
https://en.wikipedia.org/wiki/Möbius_transformation
I
s
o
m
+
(
H
3
)
=
P
G
L
(
2
,
C
)
=
P
S
L
(
2
,
C
)
{\displaystyle Isom^+(\mathbb{H}^3)=PGL(2, \mathbb{C})=PSL(2, \mathbb{C})}
+ ,
I
s
o
m
(
S
2
)
=
O
(
3
)
{\displaystyle Isom(\mathbb{S}^2)=O(3)}
+
S
2
,
E
2
,
H
2
,
S
O
(
3
)
,
I
S
O
(
R
2
)
+
,
S
L
(
2
,
R
)
=
S
O
(
1
,
3
)
+
,
π
1
(
S
2
)
=
0
,
π
1
(
T
≅
R
2
/
Z
2
)
=
Z
2
{\displaystyle \mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2,SO(3),ISO(\mathbb{R}^2)^+,SL(2,\mathbb{R})=SO(1,3)^+,\pi_1(\mathbb{S}^2)=0,\pi_1(T \cong \R^2/\Z^2)=\Z^2}
RP n
CP n
HP n
T
≅
U
(
1
)
≅
R
/
Z
≅
SO
(
2
)
.
{\displaystyle \mathbb T \cong \mbox{U}(1) \cong \mathbb R/\mathbb Z \cong \mbox{SO}(2).}
e
i
θ
=
cos
θ
+
i
sin
θ
.
{\displaystyle e^{i\theta} = \cos\theta + i\sin\theta.}
ζ
S
O
(
2
)
(
s
)
=
ζ
(
s
)
{\displaystyle \zeta_{SO(2)}(s)=\zeta(s)}
SU
(
2
)
{\displaystyle \mbox{SU}(2)}
e
i
(
θ
/
2
)
(
n
^
⋅
σ
)
=
I
2
cos
θ
/
2
+
i
(
n
^
⋅
σ
)
sin
θ
/
2
,
{\displaystyle e^{i(\theta/2)(\hat n \cdot \sigma)} = I_2 \cos \theta/2 + i(\hat n \cdot \sigma) \sin \theta/2,}
ζ
S
U
(
2
)
(
s
)
=
ζ
(
s
)
{\displaystyle \zeta_{SU(2)}(s)=\zeta(s)}
SO
(
3
)
{\displaystyle \mbox{SO}(3)}
e
i
θ
(
n
^
⋅
J
)
=
I
3
+
i
(
n
^
⋅
J
)
sin
θ
+
(
n
^
⋅
J
)
2
(
cos
θ
−
1
)
,
{\displaystyle 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),}
ζ
S
U
(
3
)
(
s
)
=
∑
x
=
1
∞
∑
y
=
1
∞
1
(
x
y
(
x
+
y
)
/
2
)
s
.
{\displaystyle \zeta_{SU(3)}(s)=\sum_{x=1}^{\infty}\sum_{y=1}^{\infty}\frac{1}{(xy(x+y)/2)^s}.}
+
+
+
+
+
+
+
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
Khatri rao product
map
equivalence
V
→
V
{\displaystyle V \to V}
M
≃
B
−
1
M
B
{\displaystyle M \simeq B^{-1} MB}
V
×
V
∗
→
R
{\displaystyle V\times V^*\to\mathbb{R}}
M
≃
B
T
M
B
{\displaystyle M \simeq B^TMB}
+
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]
spectral stuff
https://en.wikipedia.org/wiki/Singular_integral
https://en.wikipedia.org/wiki/Singular_trace
linear algebra
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
https://en.wikipedia.org/wiki/Matrix_pencil
https://en.wikipedia.org/wiki/Rouché-Capelli_theorem
stuff with det,tr
Poisson summation formula Frobenius reciprocity Selberg trace formula + + + + + Poisson=Fourier on circle
d
d
t
e
X
(
t
)
=
∫
0
1
e
α
X
(
t
)
d
X
(
t
)
d
t
e
(
1
−
α
)
X
(
t
)
d
α
=
e
X
1
−
e
−
a
d
X
a
d
X
d
X
(
t
)
d
t
.
{\displaystyle \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}.}
+
+
+
https://en.wikipedia.org/wiki/Trigonometric_functions_of_matrices
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
http://mathworld.wolfram.com/Convergent.html
simplex determinant
Cayley-Menger_determinant
a
⋅
b
=
‖
a
‖
‖
b
‖
cos
θ
,
{\displaystyle \mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\ \|\mathbf{b}\|\cos\theta ,}
ζ
(
n
+
m
)
=
ζ
(
2
n
)
ζ
(
2
m
)
cos
θ
,
{\displaystyle \zeta(n+m)=\sqrt{\zeta(2n)\zeta(2m)}\cos\theta ,}
+
+
https://en.wikipedia.org/wiki/Doubly_periodic_function
https://en.wikipedia.org/wiki/Fundamental_pair_of_periods
div
F
=
∇
⋅
F
=
tr
(
J
(
f
)
)
{\displaystyle \operatorname{div} \mathbf{F} = \nabla\cdot\mathbf{F} =\operatorname{tr}(\mathbf J(f))}
+
Δ
f
=
∇
2
f
=
∇
⋅
∇
f
=
tr
(
H
(
f
)
)
{\displaystyle \Delta f = \nabla^2 f = \nabla \cdot \nabla f =\operatorname{tr}\big(H(f)\big)}
+
+
https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials
https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case
https://en.wikipedia.org/wiki/Vanishing_theorem
https://en.wikipedia.org/wiki/Analytic_torsion
https://en.wikipedia.org/wiki/Crooks_fluctuation_theorem
E
n
=
2
n
π
k
T
=
ℏ
ω
n
{\displaystyle E_n= 2 n \pi k T=\hbar \omega n }
+
+
+
⟨
A
^
⟩
=
1
Z
0
Tr
[
ρ
0
^
A
^
]
=
1
Z
0
∑
n
⟨
n
|
A
^
|
n
⟩
e
−
β
E
n
{\displaystyle \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}}
ρ
0
^
=
e
−
β
H
^
0
=
∑
n
|
n
⟩
⟨
n
|
e
−
β
E
n
{\displaystyle \hat{\rho_0}=e^{-\beta \hat{H}_0}=\sum_n |n \rangle\langle n |e^{-\beta E_n}}
+
+
+
+
+
+
+
⟨
Γ
i
Γ
j
⟩
=
tr
{
Γ
i
Γ
j
R
0
}
{\displaystyle \langle \Gamma_i\Gamma_j \rangle=\operatorname{tr}\{\Gamma_i\Gamma_jR_0\}}
⟨
Γ
i
(
t
)
Γ
j
(
t
′
)
⟩
∝
δ
(
t
−
t
′
)
ideally
{\displaystyle \langle \Gamma_i(t)\Gamma_j(t') \rangle \propto \delta(t-t')\quad\text{ideally} }
+
|
ψ
(
t
)
⟩
=
exp
(
−
i
ℏ
H
^
t
)
|
q
0
⟩
≡
exp
(
−
i
ℏ
H
^
t
)
|
0
⟩
{\displaystyle |\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
}
⟨
F
|
exp
(
−
i
ℏ
H
^
T
)
|
0
⟩
=
∫
D
q
(
t
)
exp
[
i
ℏ
S
]
{\displaystyle \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]}
+
∫
exp
(
−
(
X
−
E
[
X
]
)
2
2
E
[
(
X
−
E
[
X
]
)
2
]
)
=
(
det
(
E
[
(
X
−
E
[
X
]
)
2
)
)
1
2
{\displaystyle \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}}}
Θ
↦
E
[
exp
(
i
tr
(
X
Θ
)
)
]
=
|
I
−
2
i
Θ
V
|
−
n
/
2
{\displaystyle \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} }
+
−
F
k
T
=
ln
Tr
exp
(
−
1
k
T
H
^
)
{\displaystyle -\frac{F}{k T} = \ln \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)}
+
Index
(
D
)
=
dim
Ker
(
D
)
−
dim
Ker
(
D
∗
)
=
tr
(
exp
(
D
∗
D
)
)
−
tr
(
exp
(
D
D
∗
)
)
{\displaystyle \operatorname{Index}(D) = \dim\operatorname{Ker}(D)− \dim\operatorname{Ker}(D*)=\operatorname{tr}(\exp(D*D))-\operatorname{tr}(\exp(DD*))}
+
1
Z
GUE
(
n
)
e
−
n
2
t
r
H
2
{\displaystyle \frac{1}{Z_{\text{GUE}(n)}} e^{- \frac{n}{2} \mathrm{tr} H^2} }
+
ζ
(
1
−
n
,
a
)
=
−
B
n
(
a
)
n
{\displaystyle \zeta(1-n,a)=-\frac{B_n(a)}{n} \!}
for
n
≥
1
{\displaystyle n\geq1 \!}
+
ζ
(
2
n
)
=
(
−
1
)
n
+
1
B
2
n
(
2
π
)
2
n
2
(
2
n
)
!
{\displaystyle \zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}}
ζ
(
−
n
)
=
(
−
1
)
n
B
n
+
1
n
+
1
{\displaystyle \zeta(-n)=(-1)^n \frac{B_{n+1}}{n+1}}
https://en.wikipedia.org/wiki/Dedekind_psi_function
https://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis
p
(
x
)
=
1
2
π
∫
R
e
i
t
x
P
(
t
)
d
t
=
1
2
π
∫
R
e
i
t
x
φ
X
(
t
)
¯
d
t
.
{\displaystyle 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.}
+
+
+
+
f
^
(
x
)
=
1
2
π
∫
−
∞
+
∞
φ
^
(
t
)
ψ
h
(
t
)
e
−
i
t
x
d
t
=
1
2
π
∫
−
∞
+
∞
1
n
∑
j
=
1
n
e
i
t
(
x
j
−
x
)
ψ
(
h
t
)
d
t
=
1
n
h
∑
j
=
1
n
1
2
π
∫
−
∞
+
∞
e
−
i
(
h
t
)
x
−
x
j
h
ψ
(
h
t
)
d
(
h
t
)
=
1
n
h
∑
j
=
1
n
K
(
x
−
x
j
h
)
{\displaystyle
\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)
}
φ
Z
n
(
t
)
=
(
φ
Y
1
(
t
n
)
)
n
=
(
e
−
1
2
(
t
n
)
2
)
n
=
e
−
t
2
2
{\displaystyle \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}}}
K
~
(
p
;
T
)
=
G
~
ε
(
p
)
T
/
ε
=
(
e
−
1
2
(
ε
p
)
2
)
T
/
ε
=
e
−
T
p
2
2
{\displaystyle
\tilde{K}(p; T)=\tilde{G}_\varepsilon(p)^{T/\varepsilon}=(e^{-\frac{1}{2}(\sqrt{\varepsilon} p)^2})^{T/\varepsilon}= e^{-\frac{T p^2}{2}}
}
ψ
t
(
y
)
=
∫
ψ
0
(
x
)
K
(
x
−
y
;
t
)
d
x
=
∫
ψ
0
(
x
)
∫
x
(
0
)
=
x
x
(
t
)
=
y
e
i
S
D
x
,
{\displaystyle \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,}
K
(
x
,
y
;
T
)
=
∫
x
(
0
)
=
x
x
(
T
)
=
y
∏
t
exp
(
−
1
2
(
x
(
t
+
ε
)
−
x
(
t
)
ε
)
2
ε
)
D
x
,
{\displaystyle K(x, y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,Dx,}
χ
{\displaystyle \chi}
V
(
G
)
−
E
(
G
)
+
F
(
G
)
=
k
+
1
{\displaystyle 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
List of fixed-point theorems +
∑
i
(
−
1
)
i
Tr
(
Frob
,
H
c
i
(
X
,
Q
ℓ
)
)
=
|
X
(
F
q
)
|
{\displaystyle \sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))=|X({\bf F}_q)|}
∑
i
(
−
1
)
i
T
r
(
f
∗
|
H
k
(
X
,
Q
)
)
=
∑
x
∈
F
i
x
(
f
)
i
n
d
e
x
x
f
{\displaystyle \sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q}))=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index} _x f}
∑
i
(
−
1
)
i
dim
H
k
(
X
,
Q
)
=
∑
x
∈
S
i
n
g
(
v
)
i
n
d
e
x
x
v
{\displaystyle \sum_i(-1)^i\dim H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv}
+
+
+
+
+
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
∑
(
−
1
)
γ
C
γ
=
χ
(
M
)
{\displaystyle \sum(-1)^\gamma C^\gamma\,=\chi(M)}
+
∑
i
index
x
i
(
v
)
=
χ
(
M
)
{\displaystyle \sum_i \operatorname{index}_{x_i}(v) = \chi(M)\,}
+
Tr
[
(
−
1
)
F
e
−
β
H
]
=
∑
p
∈
Z
(
−
1
)
p
b
p
=
χ
(
M
)
.
{\displaystyle \textrm{Tr}[(-1)^F e^{-\beta H}]=\sum_{p\in\mathbb{Z}}(-1)^pb_p=\chi(M) \ . }
+
+
W
=
Tr
(
−
1
)
n
^
⟨
M
t
′
t
∗
⟩
noise
=
⟨
Tr
(
−
1
)
n
^
M
t
′
t
∗
⟩
noise
=
I
L
{\displaystyle {\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
=
Tr
(
−
1
)
n
^
M
t
′
t
∗
=
∑
x
∈
fix
M
t
t
′
sign
det
(
δ
j
i
−
∂
M
t
t
′
i
(
x
)
/
∂
x
j
)
{\displaystyle 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
,
Ω
,
p
)
:=
∑
y
∈
f
−
1
(
p
)
sgn
det
(
1
−
D
f
(
y
)
)
{\displaystyle \deg(f,\Omega,p):=\sum_{y\in f^{-1}(p)} \operatorname{sgn} \det(1- Df(y))}
+
deg
(
f
)
=
∑
x
∈
f
−
1
(
y
)
sgn
(
det
(
d
f
x
)
)
{\displaystyle \deg(f) = \sum_{x \in f^{-1}(y)} \operatorname{sgn} (\det(df_x))}
+
σ
(
n
)
=
∑
i
∈
Z
(
−
1
)
i
+
1
(
σ
(
n
−
1
2
(
3
i
2
−
i
)
)
+
δ
(
n
,
1
2
(
3
i
2
−
i
)
)
n
)
=
σ
(
n
−
1
)
+
σ
(
n
−
2
)
−
σ
(
n
−
5
)
−
σ
(
n
−
7
)
+
σ
(
n
−
12
)
+
σ
(
n
−
15
)
+
⋯
{\displaystyle \sigma(n)=\sum_{i\in\mathbb{Z}} (-1)^{i+1}\left(\sigma(n{-}\frac12(3i^2{-}i))+\delta(n,\frac12(3i^2{-}i))\,n\right)=\sigma(n{-}1)+\sigma(n{-}2)-\sigma(n{-}5)-\sigma(n{-}7)+\sigma(n{-}12)+\sigma(n{-}15)+ \cdots}
+
Maximum modulus principle + Fundamental theorem of algebra
Hairy ball theorem real root
https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas#Combinatorics_and_topology_method
Riemann zeta function
Hurwitz zeta function
Polylogarithm
delete me?
1+2+3+4+...
Riemann+Euler-Maclaurin
Darboux's formula
Analytic torsion
Heat kernel signature
Atiyah Singer index theorem +
Signature operator
Equivariant_index theorem
Bott residue formula
+
+
+
Supersymmetric atiyah Singer index theorem
+
+?
Equivalences: Abel–Plana_formula <=>Euler–Maclaurin_formula <=>Poisson summation formula
Singular Kernel
Regularization
[
f
]
=
[
P
0
+
1
P
1
+
1
P
2
+
⋱
]
=
∑
i
≥
0
(
−
1
)
i
[
P
i
]
{\displaystyle [f]=[P_0+\frac{1}{P_1+\frac{1}{P_2+\ddots}}]=\sum_{i \geq 0} (-1)^i[P_i]}
+
+
Algebraic Geometry
https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces
https://en.wikipedia.org/wiki/Arithmetic_of_abelian_varieties
Hilbert's basis theorem
Hilbert's Nullstellensatz
Hilbert's syzygy theorem
https://en.wikipedia.org/wiki/Genus-degree_formula https://en.wikipedia.org/wiki/Plücker_formula https://en.wikipedia.org/wiki/Riemann-Hurwitz_formula
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:
+
https://mathoverflow.net/a/17139/142708
Hodge duality + ->Poincaré duality ->Grothendieck local duality ->Serre duality
poincare-serre connection
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
kunneth theorem
+
+
+
Riemann-Roch Stuff
ℓ
(
K
X
−
D
)
=
dim
H
0
(
X
,
ω
X
⊗
L
(
D
)
∨
)
,
H
0
(
X
,
ω
X
⊗
L
(
D
)
∨
)
{\displaystyle \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 +
ℓ
(
D
)
−
ℓ
(
K
−
D
)
=
deg
(
D
)
−
g
+
1
=
{\displaystyle \ell(D)-\ell(K-D) = \deg(D) - g + 1=}
dimension − correction = degree − genus + 1.
+
χ
(
D
)
=
χ
(
0
)
+
1
2
(
D
.
D
−
D
.
K
)
{\displaystyle \chi(D) = \chi(0) + \frac{1}{2}(D.D - D.K) }
+
six operations Images of sheaves
e
i
f
i
=
d
e
g
(
X
/
X
/
G
)
=
|
G
|
{\displaystyle e_i f_i= \mathrm{deg}(X/\, X/G)= |G|}
+
https://en.wikipedia.org/wiki/Category:Geometry_of_divisors
https://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence
Hodge stuff
dim
H
0
(
X
,
C
)
−
dim
H
1
(
X
,
C
)
+
dim
H
2
(
X
,
C
)
=
2
(
dim
H
0
(
X
,
O
)
−
dim
H
1
(
X
,
O
)
)
{\displaystyle \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
χ
(
M
#
N
)
=
χ
(
M
)
+
χ
(
N
)
−
χ
(
S
n
)
.
{\displaystyle \chi(M \# N) = \chi(M) + \chi(N) - \chi(S^n).}
+
π
1
(
X
×
Y
)
=
π
1
(
T
)
∗
π
i
(
S
1
)
π
1
(
T
)
{\displaystyle \pi_1(X\times Y)=\pi_1(T)\ast_{\pi_i(S^1)}\pi_1(T)}
+
⟨
a
,
b
,
c
,
d
|
a
b
c
d
=
1
⟩
{\displaystyle \langle a,b,c,d\ |\ abcd=1\rangle}
⟨
a
,
b
,
c
|
[
a
,
b
]
c
=
1
⟩
{\displaystyle \langle a,b,c\ |\ [a,b]c=1\rangle}
+
https://en.wikipedia.org/wiki/Homology_(mathematics)
NOTES:
For a non-orientable surface, a hole is equivalent to two cross-caps.
Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere
S
2
{\displaystyle 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
×
B
=
1
2
(
A
B
−
B
A
)
{\displaystyle A \times B = \tfrac{1}{2}(AB - BA) }
+
a
×
b
=
⋆
(
a
∧
b
)
.
{\displaystyle a \times b = \star (a \wedge b) \,.}
+
a
×
b
=
[
a
]
×
b
{\displaystyle \mathbf{a} \times \mathbf{b} = [\mathbf{a}]_{\times} \mathbf{b}}
+ +
sympletic geometry
∫
M
e
−
t
H
ω
n
/
n
!
=
∑
p
e
−
t
H
(
p
)
t
n
∏
α
j
(
p
)
.
{\displaystyle \int_M e^{-tH} \omega^n/n! = \sum_p {e^{-tH(p)} \over t^n \prod \alpha_j(p)}.}
+
+
+
!
Differential geo
Maxwell's equations-Alternative formulations
Mathematical descriptions of the electromagnetic field
P-form electrodynamics
https://en.wikipedia.org/wiki/Category:Theorems_in_Riemannian_geometry
https://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry
R
a
b
⏟
Ricci
≡
R
c
a
c
b
⏟
Riemann
=
g
c
d
R
c
a
d
b
⏟
Riemann
{\displaystyle \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
∫
M
K
d
A
+
∫
∂
M
k
g
d
s
=
2
π
χ
(
M
)
,
{\displaystyle \int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \, }
+
ζ
′
(
Δ
,
0
)
=
1
12
∫
M
K
d
A
{\displaystyle \zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA}
+
+
∬
R
|
N
u
×
N
v
|
d
u
d
v
=
∬
R
K
|
X
u
×
X
v
|
d
u
d
v
=
∬
S
K
d
A
{\displaystyle \iint_R |N_u \times N_v| \ du\, dv = \iint_R K|X_u \times X_v| \ du\, dv = \iint_S K \ dA}
+
R
a
b
=
K
g
a
b
.
{\displaystyle \operatorname{R}_{ab} = Kg_{ab}. \, }
+
S
=
tr
g
Ric
{\displaystyle S=\operatorname{tr}_g \operatorname{Ric}}
+
R
k
l
¯
=
∂
k
∂
l
¯
ln
(
det
(
g
)
)
,
Ricci-Chern form
{\displaystyle R_{k \overline{l}}=\partial_{k} \partial_{\overline{l}} \ln (\operatorname{det}(g)),\ \text{Ricci-Chern form}}
+
+
+
log
(
T
a
n
M
i
)
volume
(
M
i
)
→
−
1
6
π
;
{\displaystyle \frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi};}
e
x
p
(
−
ζ
′
(
0
)
)
/
V
o
l
(
S
)
{\displaystyle exp( - \zeta'(0)) / Vol(S)}
+
+
Real n-Cauchy-Riemann:
D
f
T
D
f
=
(
det
(
D
f
)
)
2
/
n
I
{\displaystyle Df^TDf = (\det(Df))^{2/n}I}
+
D
ψ
D
ψ
¯
=
∏
i
d
a
i
d
b
i
=
∏
i
d
a
′
i
d
b
′
i
det
−
2
(
C
j
i
)
,
{\displaystyle \mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^i db^i = \prod\limits_i da^{\prime i}db^{\prime i}{\det}^{-2}(C^i_j),}
+
Volume form
+
Connection_form
torsion form
curvature form
+
spin connection
khäler form
+
Solder form
Conformal connection
∫
X
ω
∧
α
=
∫
X
d
g
∧
ω
=
∫
γ
×
(
0
,
ε
)
d
g
∧
ω
=
∫
γ
×
(
0
,
ε
)
d
(
g
ω
)
=
∫
γ
ω
.
{\displaystyle \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.}
+
d
p
d
q
=
∂
(
p
,
q
)
∂
(
θ
,
φ
)
d
θ
d
φ
=
(
∂
p
∂
θ
∂
q
∂
φ
−
∂
p
∂
φ
∂
q
∂
θ
)
d
θ
d
φ
=
n
2
cos
θ
sin
θ
d
θ
d
φ
=
n
2
cos
θ
d
Ω
,
{\displaystyle \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
=
1
4
E
[
(
d
log
p
)
2
]
+
E
[
(
d
α
)
2
]
−
(
E
[
d
α
]
)
2
−
i
2
E
[
d
log
p
∧
d
α
]
=
1
4
E
[
∂
log
p
∂
θ
j
∂
log
p
∂
θ
k
]
+
E
[
∂
α
∂
θ
j
∂
α
∂
θ
k
]
−
E
[
∂
α
∂
θ
j
]
E
[
∂
α
∂
θ
k
]
−
i
2
E
[
∂
log
p
∂
θ
j
∂
α
∂
θ
k
−
∂
α
∂
θ
j
∂
log
p
∂
θ
k
]
{\displaystyle 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]}
+
(
∇
×
F
)
(
p
)
⋅
n
^
=
d
e
f
lim
A
→
0
(
1
|
A
|
∮
C
F
⋅
d
r
)
{\displaystyle (\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)}
+
∂
g
∂
z
¯
(
z
0
)
=
d
e
f
lim
r
→
0
1
2
π
i
r
2
∮
Γ
(
z
0
,
r
)
g
(
z
)
d
z
,
{\displaystyle {\frac{\partial g}{\partial \bar{z}}(z_0)}\overset{\mathrm{def}}{=}\lim_{r \to 0}\frac{1}{2\pi i r^2} \oint_{\Gamma(z_0,r)} g(z)\mathrm{d}z,}
+
Degenerancy theory
covering degenerancy
manifold degenerancy
Poincaré–Hopf theorem Hairy ball theorem
Banach fixed point theorem (existence and uniqueness)
Brouwer_fixed-point_theorem (existence)
Fixed point degree
Template:Analogous fixed-point theorems
+
+
Hall's marrriage theorem equivalences
Template:Sidebar
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
--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
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
https://en.wikipedia.org/wiki/Representation_theorem https://en.wikipedia.org/wiki/Multiplicity-one_theorem
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
e
d
l
R
(
R
/
I
)
{\displaystyle l_R(R/I^{[p^e]})=p^{ed}l_R(R/I)}
+
reg
(
I
r
)
=
r
d
{\displaystyle \operatorname{reg}(I^r)=rd}
+
group stuff
group theory: definitions basics facts non basic facts
groups: https://en.wikipedia.org/wiki/Bézout's_identity
Rings: https://en.wikipedia.org/wiki/Chinese_remainder_theorem
+
burside:+
https://en.wikipedia.org/wiki/Vantieghems_theorem
∑
i
=
1
p
−
1
i
p
−
1
≡
−
1
(
mod
p
)
,
∏
i
=
1
p
−
1
i
p
−
1
≡
1
(
mod
p
)
⇔
p
prime
{\displaystyle \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
)
!
≡
−
1
(
mod
p
)
,
⇔
p
prime
{\displaystyle (p-1)! \equiv -1 \pmod p,\Leftrightarrow p \text{ prime}}
∑
k
=
0
n
−
1
e
2
π
i
k
n
=
0
.
{\displaystyle \sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .}
+
∏
k
=
0
n
−
1
e
2
π
i
k
n
=
(
−
1
)
n
−
1
.
{\displaystyle \prod_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = (-1)^{n-1} .}
+
gcd
(
p
,
∑
i
=
1
p
−
1
i
p
−
1
)
=
1
⇔
p
prime or carmichael
{\displaystyle \gcd\left(p, \sum_{i=1}^{p-1} i^{p-1}\right)=1\Leftrightarrow p \text{ prime or carmichael}}
+
+
https://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician)
https://en.wikipedia.org/wiki/Chebotarev_theorem_on_roots_of_unity
[4]
Cayley's theorem equivalences Wagner-preston theorem + + +
https://en.wikipedia.org/wiki/Deligne-Lusztig_theory
https://en.wikipedia.org/wiki/Zappa–Szép_product
H
⊂
G
,
K
⊂
G
{\displaystyle H\subset G,K\subset G}
=\=>
H
K
⊂
G
{\displaystyle HK\subset G}
+
H
1
≅
K
1
,
H
2
≅
K
2
,
G
{\displaystyle H_1\cong K_1,H_2\cong K_2,\ G}
=\=>
H
1
K
1
≅
H
1
K
1
{\displaystyle \frac{H_1}{K_1}\cong\frac{H_1}{K_1}}
+
F
p
=
⟨
x
⟩
=\=>
F
p
2
=
⟨
x
⟩
{\displaystyle F_{p}=\langle x\rangle\text{=\=>}F_{p^2}=\langle x\rangle}
+
(
±
1
)
2
,
(
±
12
)
2
(
mod
143
)
≡
1
{\displaystyle (\pm1)^2,(\pm12)^2\pmod{143}\equiv 1}
+
G
=
⟨
x
⟩
⇒
G
/
Z
(
G
)
=
⟨
x
Z
(
G
)
⟩
{\displaystyle G=\langle x\rangle\Rightarrow G/Z(G)=\langle xZ(G)\rangle}
+
C
G
(
C
G
(
g
)
)
=
Z
(
C
G
(
g
)
)
{\displaystyle C_G(C_G(g)) = Z(C_G(g))}
+
gcd
(
|
G
|
,
n
)
|
|
{
x
∈
G
:
x
n
=
1
}
|
{\displaystyle \operatorname{gcd}(|G|,n)||\{x\in G:x^n=1\}|}
+
|
X
P
|
≡
|
X
|
mod
p
(P p-group)
{\displaystyle |X^{P}|\equiv |X| \mod p\quad \text{(P p-group)}}
+
a
p
≡
a
mod
p
(p prime)
{\displaystyle a^p \equiv a \mod p\quad \text{(p prime)}}
+ +
Thompson order formula
|
G
|
=
|
Z
(
G
)
|
+
∑
i
=
1
r
|
G
:
C
G
(
g
i
)
|
{\displaystyle |G| = |Z(G)| + \sum_{i=1}^r |G:C_G(g_i)|}
+
|
S
|
=
|
S
0
|
+
∑
i
=
1
r
|
G
|
/
|
G
i
|
{\displaystyle |S| = |S_0| + \sum_{i=1}^r |G|/|G_i|}
+
|
X
/
G
|
=
∑
x
∈
X
1
|
G
⋅
x
|
{\displaystyle |X/G|=\sum_{x \in X}\frac{1}{|G \cdot x|} }
|
X
/
G
|
|
G
|
=
∑
x
∈
X
|
G
x
|
{\displaystyle |X/G||G|=\sum_{x \in X} |G_x|}
|
X
/
G
|
|
G
|
=
∑
g
∈
G
|
X
g
|
{\displaystyle |X/G||G|=\sum_{g \in G}|X^g|}
|
X
|
/
|
G
|
=
∑
x
∈
X
/
G
1
|
G
x
|
{\displaystyle |X|/|G|=\sum_{x \in X/G} \frac{1}{|G_x|}}
+
|
X
|
=
∑
x
∈
X
/
G
|
G
⋅
x
|
{\displaystyle |X|=\sum_{x \in X/G}|G \cdot x|}
|
X
|
=
|
X
G
|
+
∑
x
∈
X
/
G
,
|
X
/
G
|
>
1
|
G
⋅
x
|
{\displaystyle |X|=|X^G|+\sum_{x \in X/G, |X/G|>1}|G \cdot x|}
∑
x
∈
π
0
(
X
)
1
|
Stab
(
x
)
|
=
|
S
|
|
G
|
,
∑
x
∈
π
0
(
X
)
1
|
Aut
(
x
)
|
{\displaystyle \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)|}}
+
+
∑
Λ
1
|
Aut
(
Λ
)
|
=
2
π
−
n
(
n
+
1
)
/
4
∏
j
=
1
n
Γ
(
j
/
2
)
∏
p
prime
2
m
p
(
f
)
{\displaystyle \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
∖
G
|
|
G
x
|
{\displaystyle |G|=|G/G_x||G_x|=|G_x \backslash G||G_x|}
+
|
G
|
=
|
G
/
H
|
|
H
|
=
|
H
∖
G
|
|
H
|
{\displaystyle |G|=|G/H||H|=|H \backslash G||H|}
+
G
/
Z
(
G
)
≅
I
n
n
(
G
)
{\displaystyle G/Z(G) \cong Inn(G)}
A
u
t
(
G
)
/
I
n
n
(
G
)
≅
O
u
t
(
G
)
{\displaystyle Aut(G)/Inn(G) \cong Out(G)}
X
g
=
g
X
g
−
1
{\displaystyle X^g = gXg^{-1}}
+
+
+
Chebotarev's density theorem [[5]
PNT Hardy-Ramanujan theorem
+
+
+
Centralizer-Normalizer
Orbit stabilizer
coset-index
https://en.wikipedia.org/wiki/Wedderburn's_little_theorem
https://en.wikipedia.org/wiki/Zappa–Szép_product
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
https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
https://en.wikipedia.org/wiki/Abelian_group#Classification
mobius table
φ
(
n
)
=
∑
d
∣
n
μ
(
d
)
⋅
n
d
{\displaystyle \varphi(n) = \sum_{d\mid n} \mu\left( d \right) \cdot \frac{n}{d} }
n
=
∑
d
∣
n
φ
(
d
)
{\displaystyle n=\sum_{d\mid n}\varphi(d)}
J
k
(
n
)
=
∑
d
|
n
μ
(
d
)
⋅
n
k
d
{\displaystyle J_k(n) = \sum_{d|n} \mu\left( d \right) \cdot \frac{n^k}{d} }
n
k
=
∑
d
|
n
J
k
(
d
)
{\displaystyle n^k=\sum_{d | n } J_k(d)}
Λ
(
n
)
=
−
∑
d
∣
n
μ
(
d
)
log
(
d
)
{\displaystyle \Lambda (n) = - \sum_{d \mid n} \mu(d) \log(d) \ }
log
(
n
)
=
∑
d
∣
n
Λ
(
d
)
{\displaystyle \log(n) = \sum_{d \mid n} \Lambda(d)}
π
0
(
x
)
=
∑
n
=
1
∞
μ
(
n
)
n
Π
0
(
x
1
/
n
)
{\displaystyle \pi_0(x) = \sum_{n=1}^\infty \frac{\mu(n)}n \Pi_0(x^{1/n})}
Π
0
(
x
)
=
∑
n
=
1
∞
1
n
π
0
(
x
1
/
n
)
{\displaystyle \Pi_0(x)=\sum_{n=1}^\infty \frac1n \pi_0(x^{1/n})}
R
(
x
)
=
∑
n
=
1
∞
μ
(
n
)
n
li
(
x
1
/
n
)
{\displaystyle \operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}(x^{1/n})}
li
(
x
)
=
∑
n
=
1
∞
R
(
x
1
/
n
)
n
{\displaystyle \operatorname{li}(x) = \sum_{n=1}^{\infty} \frac{\operatorname{R}(x^{1/n})}{n} }
P
(
s
)
=
∑
n
>
0
μ
(
n
)
log
ζ
(
n
s
)
n
{\displaystyle P(s)=\sum_{n>0} \mu(n)\frac{\log\zeta(ns)} n}
log
ζ
(
s
)
=
∑
n
>
0
P
(
n
s
)
n
{\displaystyle \log\zeta(s)=\sum_{n>0} \frac{P(ns)} n}
M
k
(
n
)
=
∑
d
|
n
μ
(
n
d
)
N
k
(
d
)
{\displaystyle M_k(n)\ =\ \sum\nolimits_{d|n} \mu(\tfrac{n}{d}) N_k(d)}
N
k
(
n
)
=
∑
d
|
n
M
k
(
d
)
{\displaystyle N_k(n)\ =\ \sum\nolimits_{d|n} M_k(d)}
ln
(
Φ
n
(
x
)
)
=
∑
d
∣
n
μ
(
n
d
)
ln
(
x
d
−
1
)
{\displaystyle \ln(\Phi_n(x))=\sum_{d\mid n}\mu (\frac{n}{d})\ln(x^d-1)}
ln
(
x
n
−
1
)
=
∑
d
∣
n
ln
(
Φ
d
(
x
)
)
{\displaystyle \ln(x^n - 1)=\sum_{d\mid n} \ln(\Phi_d(x))}
N
(
q
,
n
)
=
1
n
∑
d
∣
n
μ
(
d
)
q
n
/
d
{\displaystyle N(q,n)=\frac{1}{n} \sum_{d\mid n} \mu(d)q^{n/d}}
q
n
=
∑
d
∣
n
d
N
(
q
,
d
)
{\displaystyle q^n=\sum_{d\mid n} dN(q,d)}
ln
(
F
(
x
)
)
=
∑
k
>=
1
μ
(
x
)
k
l
o
g
G
(
x
k
)
{\displaystyle \ln(F(x))=\sum_{k>=1}\frac{\mu(x)}{k}log G(x^k) }
ln
(
G
(
x
)
)
=
∑
k
>=
1
F
(
x
k
)
k
{\displaystyle \ln(G(x))=\sum_{k>=1} \frac{F(x^k)}{k} }
+
+
+
+
+
mobius+
∑
d
∣
n
φ
(
d
)
=
n
{\displaystyle \sum_{d\mid n}\varphi(d)=n}
,
φ
(
n
)
=
∑
k
=
1
n
gcd
(
k
,
n
)
e
−
2
π
i
k
n
{\displaystyle \varphi (n) = \sum\limits_{k=1}^n \gcd(k,n) e^{-2\pi i\frac{k}{n}}}
;
∑
d
∣
n
μ
(
d
)
=
δ
n
1
{\displaystyle \sum_{d \mid n} \mu(d)=\delta_{n1}}
,
μ
(
n
)
=
∑
gcd
(
k
,
n
)
=
1
1
≤
k
≤
n
e
2
π
i
k
n
{\displaystyle \mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,\,n)=1}} e^{2\pi i \frac{k}{n}}}
Lambert
ln
(
∑
n
=
1
∞
1
n
s
)
)
=
ln
(
(
∑
n
=
1
∞
μ
(
n
)
n
s
)
−
1
)
=
ln
(
ζ
(
s
)
)
=
ln
(
∏
p
prime
(
1
−
p
−
s
)
−
1
)
=
∑
p
,
n
p
−
n
s
n
=
∑
n
=
2
∞
Λ
(
n
)
log
(
n
)
n
n
s
+
1
=
.
.
.
=
.
.
.
{\displaystyle \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}}=...=...}
ln
(
∑
k
=
0
∞
a
k
q
k
)
=
ln
(
(
∑
n
=
0
∞
p
(
k
)
q
k
)
−
1
)
=
ln
(
f
(
q
)
)
=
ln
(
∏
k
=
1
∞
(
1
−
q
k
)
)
=
−
∑
k
,
n
q
n
k
n
=
−
∑
n
σ
(
n
)
n
q
n
=
∑
n
=
1
∞
1
n
q
n
q
n
−
1
=
,
,
,
{\displaystyle \ln(\sum_{k=0}^\infty a_kq^k)=\ln((\sum_{n=0}^\infty p(k) q^k)^{-1})=\ln(f(q))=\ln(\prod_{k=1}^\infty (1-q^k))=-\sum_{k,n}\frac{q^{nk}}{n}=-\sum_n\frac{\sigma(n)}{n}q^{n}=\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{q^n-1}=,,,}
.
.
.
=
.
.
.
=
ln
(
g
(
q
)
)
=
ln
(
∏
k
≥
1
(
1
−
q
k
)
μ
(
k
)
/
k
)
=
−
∑
k
,
n
μ
(
k
)
k
q
n
k
n
=
−
∑
n
δ
n
1
n
q
n
=
−
∑
n
=
1
∞
μ
(
n
)
n
q
n
q
n
−
1
{\displaystyle ...=...=\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
(
∏
k
≥
1
(
1
−
q
k
)
φ
(
k
)
k
)
=
−
∑
k
,
n
φ
(
k
)
k
q
n
k
n
=
−
∑
n
n
n
q
n
=
−
∑
n
=
1
∞
φ
(
n
)
n
q
n
q
n
−
1
{\displaystyle ...=...=\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
(
∏
k
≥
1
(
1
−
q
k
)
Λ
(
k
)
k
)
=
−
∑
k
,
n
Λ
(
k
)
k
q
n
k
n
=
−
∑
n
ln
(
n
)
n
q
n
=
−
∑
n
=
1
∞
Λ
(
n
)
n
q
n
q
n
−
1
{\displaystyle ...=...=\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}}
∑
n
=
1
∞
s
p
t
(
n
)
q
n
=
1
(
q
)
∞
∑
n
=
1
∞
q
n
∏
m
=
1
n
−
1
(
1
−
q
m
)
1
−
q
n
{\displaystyle \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}}
+ +
η
(
τ
)
=
q
1
24
∏
n
=
1
∞
(
1
−
q
n
)
=
(
∑
n
>
0
τ
(
n
)
q
n
)
−
24
=
(
∑
n
=
0
∞
p
(
n
)
q
n
−
1
24
)
−
1
{\displaystyle \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}}
∑
n
≥
1
τ
(
n
)
q
n
=
q
∏
n
≥
1
(
1
−
q
n
)
24
=
η
(
z
)
24
=
Δ
(
z
)
,
{\displaystyle \sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} = \eta(z)^{24}=\Delta(z),}
+
+
∑
g
=
0
∞
∑
k
=
1
∞
∑
β
∈
H
2
(
M
,
Z
)
BPS
(
g
,
β
)
1
k
(
2
sin
(
k
λ
2
)
)
2
g
−
2
q
k
β
=
∑
g
=
0
∞
∑
β
∈
H
2
(
M
,
Z
)
GW
(
g
,
β
)
q
β
λ
2
g
−
2
{\displaystyle \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}}
+
elliptic stuff
∫
0
A
ω
+
∫
0
B
ω
=
∫
0
A
⊕
B
ω
{\displaystyle \int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega}
+
|
Z
/
p
Z
[
α
]
|
=
|
Z
/
p
Z
|
d
{\displaystyle |\mathbb{Z}/p\mathbb{Z}[\alpha]|=|\mathbb{Z}/p\mathbb{Z}|^d}
+
∫
γ
f
(
ζ
)
d
ζ
+
∫
τ
−
1
f
(
ζ
)
d
ζ
=
∮
γ
τ
−
1
f
(
ζ
)
d
ζ
=
0.
{\displaystyle \int_{\gamma} f(\zeta)\,d\zeta + \int_{\tau^{-1}} f(\zeta) \, d\zeta =\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta = 0.}
+
arithmetic
[
F
p
¯
:
F
p
]
=
∞
{\displaystyle [ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty}
+ +
σ
(
ζ
n
)
=
ζ
n
p
mod
n
{\displaystyle \sigma(\zeta_n) = \zeta_n^{p \mod n}}
+
Finite unramified extension
L
/
K
⇔
{\displaystyle L/K \Leftrightarrow }
Finite unramified ring extension
O
L
/
O
K
⇔
{\displaystyle \mathcal O_L / \mathcal O_K \Leftrightarrow }
Finite extension
l
/
k
{\displaystyle l/k}
+ +
[
F
p
¯
:
F
p
]
=
∞
{\displaystyle [ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty}
+
primes stuff
∫
2
Y
(
∑
2
<
p
≤
x
log
p
−
∑
2
<
n
≤
x
1
)
2
d
x
{\displaystyle \int_2^Y\left(\sum_{2<p\le x} \log p -\sum_{2<n\le x}1\right)^2\,dx}
+
1
τ
n
∑
d
∣
n
d
2
−
(
1
τ
n
∑
d
∣
n
d
)
2
{\displaystyle \frac1{\tau_n} \sum_{d\mid n} d^2 - \left(\frac1{\tau_n} \sum_{d\mid n} d\right)^2}
+
Digamma function Weierstrass zeta function
β
(
s
)
=
∏
p
≥
3
p
prime
1
1
−
(
−
1
)
p
−
1
2
p
−
s
.
{\displaystyle \beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.}
+
G
≅
G
2
⊕
⨁
p
≡
1
(
mod
4
)
G
p
.
{\displaystyle G \cong G_2 \oplus \bigoplus_{p \, \equiv \, 1 \, (\text{mod } 4)} G_p.}
+
∑
p
≤
x
1
p
=
∫
2
x
1
t
d
(
π
(
t
)
)
{\displaystyle \sum_{p\leq x}\frac{1}{p}=\int_2^x \frac{1}{t}\,d(\pi(t))}
+
1
ζ
(
s
)
=
∑
n
=
1
∞
μ
(
n
)
n
s
{\displaystyle \frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}}
ζ
(
s
)
ζ
(
2
s
)
=
∑
n
=
1
∞
|
μ
(
n
)
|
n
s
{\displaystyle \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^{s}} }
ζ
(
s
)
=
∑
n
=
1
∞
1
n
s
{\displaystyle \zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}}
+
Π
0
(
x
)
=
li
(
x
)
−
∑
ρ
li
(
x
ρ
)
−
ln
2
+
∫
x
∞
d
t
t
(
t
2
−
1
)
ln
t
.
{\displaystyle \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
ζ
(
s
)
=
s
∫
0
∞
Π
0
(
x
)
x
−
s
−
1
d
x
.
{\displaystyle \ln \zeta(s) = s\int_0^\infty \Pi_0(x)x^{-s-1}\,\mathrm{d}x. }
π
0
(
x
)
=
R
(
x
)
−
∑
ρ
R
(
x
ρ
)
−
1
ln
x
+
1
π
arctan
π
ln
x
{\displaystyle \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
ζ
(
s
)
=
s
∫
0
∞
π
(
x
)
x
(
x
s
−
1
)
d
x
{\displaystyle \ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x}
π
(
x
)
=
li
(
x
)
+
O
(
x
log
x
)
{\displaystyle \pi(x) = \operatorname{li}(x) + O\left(\sqrt x \log x\right)}
∑
n
≤
x
Λ
(
n
)
=
x
−
∑
ρ
x
ρ
ρ
−
ln
2
π
−
1
2
ln
(
1
−
x
−
2
)
=
x
−
∑
ρ
x
ρ
ρ
−
ζ
′
(
0
)
ζ
(
0
)
−
∑
k
=
1
∞
x
−
2
k
−
2
k
{\displaystyle \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
ζ
(
s
)
=
∑
n
=
2
∞
Λ
(
n
)
log
(
n
)
1
n
s
,
Re
(
s
)
>
1.
{\displaystyle \ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}, \qquad \text{Re}(s) > 1.}
ζ
(
s
)
=
1
2
(
s
−
1
)
(
π
e
γ
)
s
/
2
∏
n
=
1
∞
(
1
+
s
2
n
)
e
−
s
/
2
n
∏
ρ
(
1
−
s
ρ
)
{\displaystyle \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})}
+
+ +
ζ
(
s
)
=
s
∫
0
∞
S
(
x
)
x
−
s
−
1
d
x
=
∫
0
∞
S
′
(
x
)
x
−
s
d
x
{\displaystyle \zeta(s)=s \int_0^\infty S(x)\,x^{-s-1}dx=\int_0^\infty S'(x)\,x^{-s}dx}
ζ
′
(
s
)
=
−
s
∫
0
∞
T
(
x
)
x
−
s
−
1
d
x
=
−
∫
0
∞
T
′
(
x
)
x
−
s
d
x
{\displaystyle \zeta'(s)=-s\int_0^\infty T(x)\,x^{-s-1}dx=-\int_0^\infty T'(x)\,x^{-s}dx}
ln
ζ
(
s
)
=
s
∫
0
∞
Π
0
(
x
)
x
−
s
−
1
d
x
=
∫
0
∞
Π
0
′
(
x
)
x
−
s
d
x
{\displaystyle \ln\zeta(s)=s\int_0^\infty \Pi_0(x)\,x^{-s-1}dx=\int_0^\infty \Pi_0'(x)\,x^{-s}dx}
ζ
′
(
s
)
ζ
(
s
)
=
−
s
∫
0
∞
ψ
(
x
)
x
−
s
−
1
d
x
=
−
∫
0
∞
ψ
′
(
x
)
x
−
s
d
x
{\displaystyle \frac{\zeta'(s)}{\zeta(s)}=-s\int_0^\infty \psi(x)\,x^{-s-1}dx=-\int_0^\infty \psi'(x)\,x^{-s}dx}
1
ζ
(
s
)
=
s
∫
1
∞
M
(
x
)
x
s
+
1
d
x
{\displaystyle \frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx}
+
ψ
0
(
x
)
=
1
2
π
i
∫
σ
−
i
∞
σ
+
i
∞
(
−
ζ
′
(
s
)
ζ
(
s
)
)
x
s
s
d
s
{\displaystyle \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}
ζ
′
(
s
)
ζ
(
s
)
=
−
s
∫
1
∞
ψ
(
x
)
x
s
+
1
d
x
{\displaystyle \frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx}
+
∫
0
∞
x
s
ln
(
1
−
e
−
x
)
d
x
=
−
∫
0
∞
x
s
−
1
e
−
x
1
−
e
−
x
d
x
{\displaystyle \int_0^\infty x^{s}\ln(1-e^{-x})dx = - \int_0^\infty x^{s-1} \frac{e^{-x}}{1 -e^{-x}} dx}
∫
0
1
|
L
i
s
(
e
2
π
i
x
)
|
2
d
x
=
∑
k
≥
1
|
e
2
π
i
k
x
k
s
|
2
=
∑
k
≥
1
1
k
2
s
=
ζ
(
2
s
)
{\displaystyle \int_{0}^{1} | \mathsf{Li}_{s}(e^{2 \pi i x})|^{2} dx = \sum_{k \geq 1} \left| \frac{e^{2 \pi i k x}}{k^{s}} \right|^{2} = \sum_{k \geq 1} \frac{1}{k^{2s}} = \zeta(2s)}
+
∑
p
log
p
p
s
=
∫
1
∞
d
ϑ
(
x
)
x
s
=
s
∫
1
∞
ϑ
(
x
)
x
s
+
1
d
x
{\displaystyle \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}
¿?
∑
p
log
p
p
s
−
1
=
∫
1
∞
d
ϑ
(
x
)
x
s
=
s
∫
1
∞
ϑ
(
x
)
x
s
+
1
d
x
{\displaystyle \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
ψ
′
(
x
)
=
ln
(
x
)
Π
0
′
(
x
)
{\displaystyle \quad \psi'(x)=\ln(x)\,\Pi_0'(x)}
T
′
(
x
)
=
ln
(
x
)
S
′
(
x
)
{\displaystyle \quad T'(x)=\ln(x)\,S'(x)}
S
[
x
]
=
∑
n
=
1
⌊
x
⌋
1
=
⌊
x
⌋
{\displaystyle \quad S[x]=\sum_{n=1}^{\lfloor x\rfloor}1=\lfloor x\rfloor }
T
[
x
]
=
∑
n
=
1
⌊
x
⌋
log
n
{\displaystyle \quad T[x]=\sum_{n=1}^{\lfloor x\rfloor}\log n }
ζ
(
s
)
=
1
Γ
(
s
)
∫
0
∞
(
e
x
−
1
)
−
1
x
s
−
1
d
x
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
∑
n
>
0
e
−
n
x
d
x
{\displaystyle \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}
Γ
(
s
)
=
∫
0
∞
e
−
x
x
s
−
1
d
x
{\displaystyle \Gamma(s) = \int_0^\infty e^{-x} \,x^{s-1}\, \mathrm{d}x }
+
ζ
(
s
)
=
1
2
π
−
s
2
Γ
(
s
2
)
∫
0
∞
(
θ
(
i
x
)
−
1
)
x
s
2
−
1
d
x
,
{\displaystyle \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}
θ
(
τ
)
=
∑
n
=
−
∞
∞
e
π
i
n
2
τ
{\displaystyle \theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}}
ζ
(
s
)
=
1
2
π
−
s
2
Γ
(
s
2
)
(
1
s
−
1
−
1
s
+
1
2
∫
0
1
(
θ
(
i
x
)
−
x
−
1
2
)
x
s
2
−
1
d
x
+
1
2
∫
1
∞
(
θ
(
i
x
)
−
1
)
x
s
2
−
1
d
x
)
{\displaystyle \zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}(\frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(ix)-x^{-\frac12}\right)x^{\frac{s}{2}-1}\,\mathrm{d}x + \frac{1}{2}\int_1^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x)}
+
γ
=
lim
n
→
∞
(
ln
n
−
∑
p
≤
n
ln
p
p
−
1
)
=
lim
n
→
∞
(
−
ln
n
+
∑
k
=
1
n
1
k
)
=
∫
1
∞
(
−
1
x
+
1
⌊
x
⌋
)
d
x
.
{\displaystyle \gamma = \lim_{n\to\infty}\left(\ln n - \sum_{p\le n}\frac{\ln p}{p-1}\right)= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.}
ζ
′
(
s
)
ζ
(
s
)
=
−
∑
n
=
1
∞
Λ
(
n
)
n
s
=
−
∑
p
∈
P
log
(
p
)
p
s
−
1
=
P
p
−
1
′
(
s
)
{\displaystyle \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
,
χ
)
=
s
∫
1
∞
A
(
x
)
x
s
+
1
d
x
A
(
x
)
=
∑
n
≤
x
χ
(
n
)
{\displaystyle L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx\quad A(x)=\sum_{n\le x} \chi(n)}
ζ
(
s
)
=
s
∫
1
∞
⌊
x
⌋
x
s
+
1
d
x
{\displaystyle \zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx}
+
ζ
(
s
)
=
s
s
−
1
−
s
∫
1
∞
{
x
}
x
−
s
−
1
d
x
{\displaystyle \zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} dx}
+
+
ζ
′
(
s
)
=
−
∑
n
=
2
∞
ln
(
n
)
n
s
{\displaystyle \zeta'(s) = -\sum_{n \mathop = 2}^\infty \frac{\ln \left({n}\right)}{n^s}}
+
(
ζ
′
(
s
)
ζ
(
s
)
)
2
=
∑
n
=
1
∞
∑
d
|
n
Λ
(
d
)
Λ
(
n
/
d
)
n
s
{\displaystyle \left(\frac{\zeta'(s)}{\zeta(s)}\right)^2 = \sum_{n=1}^\infty \sum_{d|n} \frac{\Lambda(d) \Lambda(n/d)}{ n^{s}}}
+
d
d
s
(
ζ
(
k
)
(
s
)
ζ
(
s
)
)
=
ζ
(
k
+
1
)
(
s
)
ζ
(
s
)
−
ζ
′
(
s
)
ζ
(
s
)
ζ
(
k
)
(
s
)
ζ
(
s
)
{\displaystyle \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)}}
+
β
(
s
)
=
∏
p
≥
3
p
prime
1
1
−
(
−
1
)
p
−
1
2
p
−
s
.
{\displaystyle \beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.}
+
ω
p
=
(
x
+
y
i
)
p
≡
x
p
+
y
p
i
p
≡
x
+
(
−
1
)
p
−
1
2
y
i
(
mod
p
)
,
{\displaystyle \omega^p = (x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p},}
+
π
k
(
x
)
∼
x
(
log
log
x
)
k
−
1
(
k
−
1
)
!
log
x
(
1
)
{\displaystyle \pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\qquad\qquad(1)}
+
π
2
(
x
)
∼
1.32
x
(
log
x
)
2
{\displaystyle \pi_2(x) \sim 1.32 \frac {x}{(\log x)^2}}
+
p
n
≈
n
log
n
+
n
(
log
log
n
−
1
)
,
{\displaystyle p_n\approx n\log n + n(\log \log n - 1),}
∑
n
≤
x
σ
0
(
n
)
≈
x
log
x
+
x
(
2
γ
−
1
)
{\displaystyle \sum_{n\le x} \sigma_0(n)\approx x\log x + x(2\gamma-1)}
lim
n
→
∞
1
log
n
∏
p
≤
n
p
p
−
1
=
e
γ
,
{\displaystyle \lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,}
lim sup
n
→
∞
σ
(
n
)
n
log
log
n
=
e
γ
{\displaystyle \limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma}
+
+
+
lim
inf
φ
(
n
)
n
log
log
n
=
e
−
γ
.
{\displaystyle \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
(
1
p
−
1
q
)
∏
n
,
m
=
1
∞
(
1
−
p
n
q
m
)
c
n
m
=
j
(
p
)
−
j
(
q
)
{\displaystyle \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}=j(p)-j(q)}
+
1
1
−
x
=
∏
n
≥
0
(
1
+
x
2
n
)
{\displaystyle \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
1
1
−
x
=
∏
i
=
0
(
1
+
x
1
×
b
i
+
x
2
×
b
i
+
x
3
×
b
i
+
.
.
.
)
{\displaystyle \frac{1}{1-x}=\prod_{i=0}(1+x^{1\times b^i}+x^{2\times b^i}+x^{3\times b^i}+...)}
∏
n
≥
0
1
1
−
x
2
n
+
1
=
∏
n
≥
0
(
1
+
x
n
)
{\displaystyle \prod_{n\geq 0}\frac{1}{1-x^{2n+1}}=\prod_{n\geq 0} (1+x^{n})}
+ +
∏
i
,
j
(
1
−
x
i
y
j
)
−
1
=
∑
λ
m
λ
(
x
)
h
λ
(
y
)
=
∑
λ
s
λ
(
x
)
s
λ
(
y
)
{\displaystyle \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
χ
(
M
g
)
=
ζ
(
1
−
2
g
)
/
(
2
−
2
g
)
{\displaystyle \chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)}
+
+ +
height stuff
max
(
|
A
+
A
|
,
|
A
⋅
A
|
)
≥
c
⋅
|
A
|
1
+
ε
{\displaystyle \max( |A+A|, |A \cdot A| ) \geq c \cdot |A|^{1+\varepsilon} }
max
(
|
a
|
,
|
b
|
,
|
c
|
)
≥
C
⋅
r
a
d
(
a
b
c
)
1
+
ε
{\displaystyle \max(|a|,|b|,|c|) \geq C \cdot rad(abc)^{1+\varepsilon} }
+
max
{
deg
(
a
)
,
deg
(
b
)
,
deg
(
c
)
}
≤
deg
(
rad
(
a
b
c
)
)
−
1.
{\displaystyle \max\{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc))-1.}
max
{
|
x
|
,
|
y
|
,
|
z
|
}
<
C
|
ξ
−
α
|
−
1
3
max
(
1
,
ξ
2
)
{\displaystyle \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
+
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
+
+
(
x
−
α
)
(
x
−
α
′
)
=
x
2
+
x
+
k
Euler-Heegner polynomial
⟺
Z
[
α
]
=
Z
[
1
+
1
−
4
k
2
]
PID
{\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 formula Shimura correspondence
Q
∈
E
(
Q
)
⇒
N
a
g
e
l
−
L
u
t
z
Q
∈
E
(
Z
)
{\displaystyle Q\in E(\Bbb Q)\Rightarrow_{Nagel-Lutz} Q\in E(\Bbb Z)}
+ +
ζ
(
s
)
=
π
s
2
Γ
(
s
2
)
∫
0
∞
v
s
2
(
θ
(
i
v
)
−
1
2
)
d
v
v
{\displaystyle \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}}}
+
ζ
(
s
)
=
∑
n
=
1
∞
1
n
s
{\displaystyle \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}}
ζ
(
s
)
=
∏
p
prime
1
1
−
p
−
s
{\displaystyle \zeta(s)=\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}}
lim
s
→
1
(
s
−
1
)
ζ
(
s
)
=
1
{\displaystyle \lim_{s\to 1}(s-1)\zeta(s)=1}
L
(
s
,
χ
)
=
∑
n
=
1
∞
χ
(
n
)
n
s
{\displaystyle L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}}
L
D
(
s
)
:=
∏
p
p
r
i
m
e
1
1
−
χ
D
(
p
)
p
−
s
{\displaystyle {L_D}(s)\!\!:=\mathop {\prod}\limits_{p\,{\rm{prime}}} {\frac{1}{{1 - {\chi _D}(p){p^{ - s}}}}} }
|
D
|
2
π
L
D
(
1
)
=
h
(
D
)
2
{\displaystyle \frac{{\sqrt {\left| D \right|} }}{{2\pi }}{L_D}(1)=\frac{{h(D)}}{2}}
Q
(
x
,
y
)
=
A
x
2
+
B
x
y
+
C
y
2
,
D
=
B
2
−
4
A
C
{\displaystyle Q(x, y)=Ax^2+Bxy+Cy^2,D=B^{2}-4AC}
Q
(
x
,
y
)
=
Q
(
a
x
+
b
y
,
c
x
+
d
y
)
,
a
d
−
b
c
=
1
{\displaystyle Q(x, y)=Q(ax+by,cx+dy),ad-bc=1}
L
E
(
s
)
=
(
2
π
)
s
Γ
(
s
)
∫
0
∞
v
s
f
E
(
i
v
)
d
v
v
{\displaystyle {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
)
:=
∏
p
p
r
i
m
e
1
1
−
a
p
p
−
s
+
ε
(
p
)
p
1
−
2
s
{\displaystyle {L_E}(s)\!\!:=\mathop{\prod}\limits_{p\,{\rm{prime}}}{\frac{1}{{1-{a_p}{p^{-s}}+\varepsilon (p){p^{1-2s}}}}}}
l
i
m
s
→
1
(
s
−
1
)
−
r
E
L
E
(
s
)
<
∞
{\displaystyle {\rm{li}}{{\rm{m}}_{s \to 1}}{(s-1)^{-{r_E}}}{L_E}(s)<\infty}
lim
s
→
1
1
Ω
E
L
E
(
s
)
(
s
−
1
)
r
E
=
c
E
|
Ш
(
E
)
|
{\displaystyle \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
+
A
x
+
B
,
4
A
3
+
27
B
2
≠
0
{\displaystyle {y^2}={x^3}+Ax+B,4A^3+27B^2\neq0}
(
N
c
z
+
d
)
−
2
f
E
(
a
z
+
b
N
c
z
+
d
)
=
f
E
(
z
)
,
a
d
−
N
b
c
=
1
{\displaystyle {(Ncz + d)^{ - 2}}{f_E}\left({\frac{{az + b}}{{Ncz + d}}}\right)={f_E}(z),ad-Nbc=1}
ζ
K
(
s
)
=
∑
I
⊆
O
K
1
(
N
K
/
Q
(
I
)
)
s
{\displaystyle \zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}}
ζ
K
(
s
)
=
∏
P
⊆
O
K
1
1
−
(
N
K
/
Q
(
P
)
)
−
s
,
for Re
(
s
)
>
1.
{\displaystyle \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
→
1
(
s
−
1
)
ζ
K
(
s
)
=
2
r
1
⋅
(
2
π
)
r
2
⋅
Reg
K
⋅
h
K
w
K
⋅
|
D
K
|
{\displaystyle \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
)
=
∏
p
∣
Δ
(
1
−
a
p
p
−
s
)
−
1
⋅
∏
p
∤
Δ
(
1
−
a
p
p
−
s
+
p
1
−
2
s
)
−
1
=
∑
n
=
1
∞
a
n
n
s
{\displaystyle 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 theorem Max-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
π
n
,
a
(
n
)
∼
π
(
n
)
φ
(
n
)
{\displaystyle \pi_{n,a}(n) \sim \frac{\pi(n)}{\varphi(n)}}
π
q
2
,
1
(
n
)
≈
π
(
n
)
q
⋅
(
q
−
1
)
{\displaystyle \pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}}
+
Artin's conjecture
+
Π
∗
(
x
;
q
,
a
)
=
∑
p
≤
x
,
p
≡
a
(
mod
q
)
∗
1
+
∑
p
2
≤
x
,
p
2
≡
a
(
mod
q
)
∗
1
2
+
∑
p
3
≤
x
,
p
3
≡
a
(
mod
q
)
∗
1
3
+
⋯
=
π
∗
(
x
;
q
,
a
)
+
1
2
∑
b
(
mod
q
)
,
b
2
≡
a
(
mod
q
)
π
∗
(
x
1
/
2
;
q
,
b
)
+
1
3
∑
c
(
mod
q
)
,
c
3
≡
a
(
mod
q
)
π
∗
(
x
1
/
3
;
q
,
c
)
+
⋯
{\displaystyle
\Pi^*(x;q,a) = \sum^*_{p\le x, p\equiv a\pmod q} 1 + \sum^*_{p^2\le x, p^2\equiv a\pmod q} \tfrac12 + \sum^*_{p^3\le x, p^3\equiv a\pmod q} \tfrac13 + \cdots
= \pi^*(x;q,a) + \tfrac12 \sum_{b\pmod q, b^2\equiv a\pmod q} \pi^*(x^{1/2};q,b) + \tfrac13 \sum_{c\pmod q, c^3\equiv a\pmod q} \pi^*(x^{1/3};q,c) + \cdots
}
+
k-tuple conjecture +
Bateman-Horn conjecture +
∑
0
<
γ
,
γ
′
<
T
,
0
<
γ
−
γ
′
<
2
π
α
ln
T
1
=
T
ln
T
2
π
(
1
+
o
(
1
)
)
∫
0
α
1
−
(
sin
(
π
u
)
π
u
)
2
d
u
.
{\displaystyle \sum_{0<\gamma,\gamma'<T, 0<\gamma-\gamma'< \frac{2\pi\alpha}{\ln T}}1=\frac{T\ln T}{2\pi}(1+o(1))\int_0^\alpha 1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2 \mathrm du.}
+
https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture
https://en.wikipedia.org/wiki/Sato-Tate_conjecture
+
+
square free distribution
+
hypergeometric stuff
Painlevé's conditions Malmquist's conditions + +
???????????????
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
Poincaré series
Igusa zeta function
Hyperbolic geometry Hyperbolic 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
Template:Commutative ring classes
Template:Hidden begin
Template:Algebraic structures
Template:Hidden end
eta stuff
Eta invariant
Dirichlet_eta_function
Spectral asymmetry
Witten index
η
(
M
)
=
1
π
∫
0
∞
t
1
/
2
T
r
[
D
exp
(
−
t
D
2
)
]
d
t
{\displaystyle \eta(M)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}t^{1/2}\mathrm{Tr}[D\exp(-tD^2)]dt}
+
1
4
(
η
L
−
(
A
)
−
η
L
−
(
0
)
)
=
c
2
(
G
)
2
π
I
[
A
]
{\displaystyle \frac{1}{4}\left(\eta_{L_{-}}(A)-\eta_{L_{-}}(0)\right)=\frac{c_{2}(G)}{2\pi}I[A]}
+
heat table
exp
(
−
c
b
2
2
)
=
∫
d
x
2
c
π
exp
(
−
x
2
2
c
±
i
b
x
)
{\displaystyle \exp(-\frac{cb^2}{2})=\int \frac{dx}{\sqrt{2c\pi}} \exp(-\frac{x^{2}}{2c} \pm ib x)}
e
−
1
2
⟨
a
|
E
|
a
⟩
∝
∫
D
ϕ
e
−
1
2
⟨
ϕ
|
E
−
1
|
ϕ
⟩
e
i
⟨
a
|
ϕ
⟩
{\displaystyle 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
(
c
b
2
2
)
=
∫
d
x
2
c
π
exp
(
−
x
2
2
c
±
b
x
)
{\displaystyle \exp(\frac{cb^2}{2})=\int \frac{dx}{\sqrt{2c\pi}} \exp(-\frac{x^{2}}{2c} \pm b x)}
e
1
2
⟨
a
|
E
|
a
⟩
∝
∫
D
ϕ
e
−
1
2
⟨
ϕ
|
E
−
1
|
ϕ
⟩
e
⟨
a
|
ϕ
⟩
{\displaystyle 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
)
=
1
4
π
∫
−
∞
∞
e
−
y
2
/
4
e
−
y
D
f
(
x
)
d
y
=
1
4
π
∫
−
∞
∞
e
−
y
2
/
4
f
(
x
−
y
)
d
y
{\displaystyle 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
(
−
c
|
b
|
2
2
)
=
∫
d
z
d
z
¯
2
i
π
a
exp
(
−
|
z
|
2
2
c
±
i
b
¯
z
±
i
b
z
¯
)
{\displaystyle \exp(-\frac{c|b|^2}{2})=\int\frac{dz d\bar{z}}{2i\pi a}\exp(-\frac{|z|^2}{2c}\pm i\overline{b}z\pm i b\overline{z})}
exp
(
c
|
b
|
2
2
)
=
∫
d
z
d
z
¯
2
i
π
a
exp
(
−
|
z
|
2
2
c
±
b
¯
z
±
b
z
¯
)
{\displaystyle \exp(\frac{c|b|^2}{2})=\int\frac{dz d\bar{z}}{2i\pi a}\exp(-\frac{|z|^2}{2c}\pm \overline{b}z\pm b\overline{z})}
+
+
+
q
q
q
H
(
Q
,
P
,
t
)
=
∑
s
=
0
∞
K
(
P
,
Q
,
s
)
s
!
(
−
t
)
s
?
{\displaystyle H(Q,P,t)=\sum_{s=0}^\infty \frac{K(P,Q,s)}{s!}(-t)^s \! \quad ?}
K
(
P
,
Q
,
s
)
=
1
Γ
(
s
)
∫
0
∞
H
(
P
,
Q
,
t
)
t
s
−
1
d
t
{\displaystyle K(P,Q,s)= \frac{1}{\Gamma(s)} \int_0^\infty H(P,Q,t) t^{s-1} dt }
K
∝
(
∫
V
D
ϕ
e
−
⟨
ϕ
,
K
ϕ
⟩
)
−
2
=
det
(
S
)
−
2
=
∑
n
=
0
∞
(
−
i
)
n
n
!
(
∏
k
=
1
n
∫
t
0
t
d
t
k
)
T
{
∏
k
=
1
n
e
i
H
0
t
k
V
e
−
i
H
0
t
k
}
.
{\displaystyle 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
)
=
∑
n
=
1
∞
f
n
(
P
)
f
n
(
Q
)
e
−
λ
n
t
{\displaystyle H(Q,P,t)=\sum_{n=1}^{\infty} f_n(P)f_n(Q) e^{- \lambda_{n}t}}
K
(
P
,
Q
,
s
)
=
∑
n
=
1
∞
f
n
(
P
)
f
n
(
Q
)
λ
n
s
{\displaystyle K(P,Q,s)=\sum_{n=1}^{\infty} \frac{f_n(P)f_n(Q)}{ \lambda_{n}^s}}
tr
H
(
Q
,
P
,
t
)
=
∫
M
H
(
P
,
P
,
t
)
d
P
=
H
(
s
)
=
tr
e
−
t
H
=
∑
i
=
1
∞
e
−
λ
i
t
{\displaystyle \operatorname{tr} H(Q,P,t)=\int_M H(P,P,t)dP=H(s)=\operatorname{tr} e^{-tH} = \sum^\infty_{i=1}e^{-\lambda_i t}}
tr
K
(
Q
,
P
,
s
)
=
∫
M
K
(
P
,
P
,
s
)
d
P
=
K
(
s
)
=
tr
K
−
s
=
∑
λ
i
λ
i
−
s
=
ζ
K
(
s
)
{\displaystyle \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
−
tr
K
′
(
0
)
=
e
−
(
tr
K
−
s
)
′
|
s
=
0
=
e
−
(
∑
λ
i
λ
i
−
s
)
′
|
s
=
0
=
e
−
∑
λ
i
ln
(
λ
i
)
λ
i
−
s
|
s
=
0
=
e
−
ζ
K
′
(
0
)
{\displaystyle \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
−
λ
K
)
=
exp
(
tr
(
ln
(
I
−
λ
K
)
)
)
=
exp
(
−
∑
n
=
1
∞
Tr
(
K
n
)
n
λ
n
)
=
∑
n
=
0
∞
(
−
λ
)
n
Tr
Λ
n
(
K
)
=
1
ζ
K
(
λ
)
{\displaystyle \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
)
=
1
(
4
π
t
)
d
/
2
e
−
|
x
−
y
|
2
/
4
t
{\displaystyle K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}\,}
H
(
t
)
∼
(
4
π
t
)
−
n
/
2
∑
m
=
0
∞
a
m
t
m
{\displaystyle H(t)\sim(4\pi t)^{-n/2}\sum^\infty_{m=0}a_m t^m }
+
+
+
+
+
+
+
+
ζ
Δ
(
s
)
=
1
Γ
(
s
)
∫
0
∞
t
s
−
1
∑
λ
∈
S
p
(
Δ
)
e
−
λ
t
d
t
=
1
Γ
(
2
s
)
∫
0
∞
t
2
s
−
1
∑
λ
∈
S
p
(
Δ
)
e
−
λ
t
d
t
.
{\displaystyle \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.}
+
ζ
(
k
)
=
∑
n
=
1
∞
1
n
k
=
∑
n
=
1
∞
∫
−
∞
∞
⋯
∫
−
∞
∞
e
−
π
n
2
(
x
1
2
+
⋯
+
x
k
2
)
d
x
1
⋯
d
x
k
=
∑
n
=
1
∞
(
1
k
!
∫
−
∞
∞
⋯
)
−
k
{\displaystyle \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}}
?+ +
ζ
′
(
Δ
,
0
)
=
1
12
∫
M
K
d
A
{\displaystyle \zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA}
+
+
n
!
=
exp
(
ln
(
n
!
)
)
=
exp
(
∑
n
ln
n
)
=
exp
(
∑
n
ln
n
n
s
|
s
=
0
)
=
exp
(
−
ζ
′
(
0
)
)
)
{\displaystyle 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
(
∑
p
ln
p
)
=
exp
(
∑
p
ln
p
p
s
|
s
=
0
)
=
exp
(
−
P
′
(
0
)
)
)
{\displaystyle p\#=\exp(\ln(p\#))=\exp(\sum_p\ln p)=\exp(\sum_p\frac{\ln p}{p^s}|_{s=0})=\exp(-P'(0)))}
p
∞
#
=
(
2
π
)
2
{\displaystyle p_{\infty}\#=(2\pi)^2}
+
∞
!
=
(
2
π
)
{\displaystyle \infty!=\sqrt(2\pi)}
+
ln
(
n
!
)
≈
n
ln
(
n
)
{\displaystyle \ln(n!)\approx n\ln(n)}
+
ln
(
p
n
#
)
≈
p
n
{\displaystyle \ln(p_n\#)\approx p_n}
+
ζ
(
s
)
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
∑
n
>
0
e
−
n
x
d
x
{\displaystyle \zeta(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}dx}
P
(
s
)
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
∑
p
>
0
e
−
p
x
d
x
{\displaystyle P(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{p>0}e^{-px}dx}
ζ
(
s
)
=
exp
(
−
∑
p
ln
(
1
−
p
−
s
)
)
=
exp
(
∑
p
,
n
p
−
n
s
n
)
{\displaystyle \zeta(s)=\exp(-\sum_p\ln(1-p^{-s}))=\exp(\sum_{p,n}\frac{p^{-ns}}{n})}
+
ϕ
(
q
)
=
exp
(
−
∑
k
ln
(
1
−
q
k
)
)
=
exp
(
∑
k
,
n
q
−
n
k
n
)
=
exp
(
∑
n
=
1
∞
1
n
q
n
q
n
−
1
)
{\displaystyle \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})}
+
ϕ
(
q
)
=
exp
(
−
∑
k
ln
(
1
−
q
k
)
)
=
exp
(
∑
k
,
n
q
−
n
k
n
)
=
exp
(
∑
k
|
m
,
m
k
q
−
m
m
)
=
exp
(
∑
m
q
−
m
m
∑
k
|
m
k
)
=
exp
(
∑
m
q
−
m
m
σ
(
m
)
)
{\displaystyle \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
)
=
∑
x
∈
X
−
log
(
1
−
T
deg
(
x
)
)
=
∑
x
∈
X
∑
n
=
1
∞
T
deg
(
x
)
⋅
n
n
=
∑
m
=
1
∞
(
∑
deg
(
x
)
|
m
deg
(
x
)
)
T
m
m
=
∑
m
=
1
∞
|
X
(
F
q
m
)
|
T
m
m
{\displaystyle \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}}
+
+
T
r
V
q
L
0
=
∑
n
∈
Z
dim
V
n
q
n
=
∏
n
≥
1
(
1
−
q
n
)
−
1
{\displaystyle Tr_V q^{L_0} = \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}}
+
℘
(
z
;
Λ
)
=
−
ζ
′
(
z
;
Λ
)
=
ln
″
(
σ
(
z
;
Λ
)
)
,
for any
z
∈
C
{\displaystyle \wp(z;\Lambda)= -\zeta'(z;\Lambda)=\ln''(\sigma(z;\Lambda)), \mbox{ for any } z \in \Complex }
+
1
2
π
i
d
d
z
log
Δ
(
z
)
=
1
−
24
∑
n
=
1
∞
n
e
2
π
i
n
z
1
−
e
2
π
i
n
z
=
1
−
24
∑
m
=
1
∞
σ
1
(
m
)
e
2
π
i
m
z
=
1
−
24
∑
n
>
0
σ
1
(
n
)
q
n
=
E
2
(
z
)
{\displaystyle \frac{1}{2 \pi i} \frac{d}{d z} \log \Delta(z)=1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n z}}{1-e^{2 \pi i n z}}=1-24 \sum_{m=1}^{\infty} \sigma_{1}(m) e^{2 \pi i m z}=1-24\sum_{n>0}\sigma_1(n)q^n=E_{2}(z)}
+
+
https://en.wikipedia.org/wiki/Hecke_operator
https://en.wikipedia.org/wiki/Witt_vector
∑
n
≤
x
Λ
(
n
)
=
1
2
π
i
∫
σ
−
i
∞
σ
+
i
∞
(
−
ζ
′
(
s
)
ζ
(
s
)
)
x
s
s
d
s
=
x
−
∑
ρ
x
ρ
ρ
−
ln
2
π
−
1
2
ln
(
1
−
x
−
2
)
=
x
−
∑
ρ
x
ρ
ρ
−
ζ
′
(
0
)
ζ
(
0
)
−
1
2
∑
k
=
1
∞
x
−
2
k
−
2
k
{\displaystyle \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
L
(
μ
,
Σ
)
=
−
n
2
ln
det
(
Σ
)
−
1
2
tr
[
Σ
−
1
∑
i
=
1
n
(
x
i
−
μ
)
(
x
i
−
μ
)
T
]
.
{\displaystyle \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]. }
∑
ρ
F
(
ρ
)
=
Tr
(
F
(
T
^
)
)
.
{\displaystyle \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 theorem Hopf–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
+
Continuously differentiable
⊆
{\displaystyle \subseteq}
Lipschitz continuous
⊆
{\displaystyle \subseteq}
α-Hölder continuous
⊆
{\displaystyle \subseteq}
uniformly continuous
⊆
{\displaystyle \subseteq}
Continuous function=continuous
0
<
α
≤
1
⇒
{\displaystyle 0 < α ≤ 1\Rightarrow}
Lipschitz continuous
⊆
{\displaystyle \subseteq}
absolutely continuous
⊆
{\displaystyle \subseteq}
bounded variation
⊆
{\displaystyle \subseteq}
differentiable
⊆
{\displaystyle \subseteq}
almost everywhere
+
https://en.wikipedia.org/wiki/Modes_of_convergence_(annotated_index)
+
LLN,LIL,CLT
LIL random matrix
PNT,RH,EK
p
n
≈
n
log
(
n
)
{\displaystyle p_n \approx n\log(n)}
|
γ
n
|
≈
2
π
n
ln
n
{\displaystyle |\gamma_n| \approx \frac{2\pi n }{\ln n}}
π
(
x
)
≈
n
log
n
{\displaystyle \pi(x)\approx \frac{n}{\log n}}
N
(
T
)
≈
T
2
π
ln
(
T
2
π
e
)
{\displaystyle 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
Selberg class
Abstract analytic number theory
PNT
RH
ζ
(
1
+
i
t
)
≠
0
{\displaystyle \zeta(1 + it)\neq 0}
ζ
(
1
/
2
+
i
t
)
=
0
{\displaystyle \zeta(1/2 + it)=0}
1
1
2
log
log
T
log
|
ζ
(
1
/
2
+
i
t
)
|
∼
N
(
0
,
1
)
{\displaystyle \frac{1}{\sqrt{\frac{1}{2} \log \log T}} \log |\zeta(1/2+it)|\sim N(0,1)}
+
π
(
x
)
=
Li
(
x
)
+
o
(
x
)
{\displaystyle \pi(x) = \operatorname{Li} (x) + o(x)}
π
(
x
)
=
Li
(
x
)
+
O
(
x
log
x
)
{\displaystyle \pi(x) = \operatorname{Li} (x) + O(\sqrt x \log x)}
∑
n
≤
x
μ
(
n
)
=
o
(
x
)
{\displaystyle \sum_{n \leq x} \mu(n) = o(x)}
∑
n
≤
x
μ
(
n
)
=
O
(
x
1
/
2
+
o
(
1
)
)
{\displaystyle \sum_{n \leq x} \mu(n) = O(x^{1/2+o(1)})}
+
1
T
μ
(
t
≤
T
|
arg
(
ζ
(
1
/
2
+
i
t
)
/
1
/
2
log
log
t
<
x
)
=
lim
T
→
∞
T
−
1
∫
0
T
1
arg
(
ζ
(
1
/
2
+
i
t
)
/
1
/
2
log
log
t
⩽
x
d
t
=
1
2
π
∫
−
∞
x
e
−
z
2
/
2
d
z
=
P
(
Z
⩽
x
)
{\displaystyle \frac{1}{T}\mu(t\le T\,|\,\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}<x)=\lim\limits_{T\to\infty}T^{-1}\int_0^T\mathbf 1_{\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}\leqslant x}\,\mathrm dt=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm e^{-z^2/2}\mathrm dz=\mathbb P(Z\leqslant x) }
+
+
One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem.
PNT scaled models +
+
+
mertens=/=PNT +
+
CLT
Martingale CLT
Functional integral Path integral