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://wikimedia.org/api/rest_v1/media/math/render/svg/6119672cd357f6618f91330f2bb194a7893d5c7d)
