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π
(
x
)
=
1
2
π
i
(
∫
a
−
∞
i
a
+
∞
i
log
ζ
(
s
)
s
∑
n
=
1
∞
μ
(
n
)
x
s
/
n
n
d
s
)
{\displaystyle \pi (x)={\frac {1}{2\pi i}}\left(\int _{a-\infty i}^{a+\infty i}{\frac {\log \zeta (s)}{s}}\sum _{n=1}^{\infty }{\frac {\mu (n)x^{s/n}}{n}}\mathrm {d} s\right)}
∫
0
a
(
x
2
n
−
1
(
a
−
x
)
)
1
2
n
b
−
x
d
x
=
2
b
π
sin
(
π
2
n
)
1
−
cos
(
π
n
)
(
1
−
a
2
b
n
−
(
1
−
a
b
)
1
2
n
)
{\displaystyle \int _{0}^{a}{\frac {\left(x^{2n-1}\left(a-x\right)\right)^{\frac {1}{2n}}}{b-x}}\mathrm {d} x={\frac {2b\pi \sin \left({\frac {\pi }{2n}}\right)}{1-\cos \left({\frac {\pi }{n}}\right)}}\left(1-{\frac {a}{2bn}}-\left(1-{\frac {a}{b}}\right)^{\frac {1}{2n}}\right)}
A
⊨
A
∨
¬
A
{\displaystyle A\models A\lor \neg A}
Ψ
=
∫
e
i
ℏ
∫
(
R
16
π
G
−
F
2
+
ψ
¯
i
D
ψ
−
λ
φ
ψ
¯
ψ
+
|
D
φ
|
2
−
V
(
φ
)
)
{\displaystyle \Psi \,=\,\int {e^{{\frac {i}{\hbar }}\int {({\frac {R}{16\pi {G}}}\,-\,F^{2}\,+\,{\overline {\psi }}iD\psi \,-\,\lambda \varphi {\overline {\psi }}\psi \,+\,|D\varphi |^{2}\,-\,V(\varphi ))}}}}
A
∼
∑
e
i
S
[
g
]
/
ℏ
{\displaystyle A\sim \sum e^{iS[g]/\hbar }}
∫
1
∞
∑
n
≤
x
∑
d
|
n
1
x
3
d
x
=
π
4
72
{\displaystyle \int _{1}^{\infty }{\frac {\sum _{n\leq x}\sum _{d|n}1}{x^{3}}}dx={\frac {\pi ^{4}}{72}}}
d
s
2
=
η
μ
υ
d
x
μ
d
x
υ
{\displaystyle ds^{2}=\eta _{\mu \upsilon }dx^{\mu }dx^{\upsilon }}
∑
n
is squarefree
1
n
{\displaystyle \sum _{n{\text{ is squarefree}}}{\frac {1}{n}}}
![{\displaystyle A\sim \sum e^{iS[g]/\hbar }}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6119672cd357f6618f91330f2bb194a7893d5c7d)
