Участник:Green fr/Простые числа

${\displaystyle p(n)=1+\left(\sum _{m=1}^{2^{n}}\left[\left[{\frac {n}{1+\left(\sum _{j=2}^{m}\left[{\frac {(j-1)!+1}{j}}-\left[{\frac {(j-1)!}{j}}\right]\right]\right)}})^{\left({\frac {1}{n}}\right)}\right]\right]\right)}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p}{\frac {1}{1-{\frac {1}{p^{s}}}}}}$

${\displaystyle {\frac {1}{1-p^{-s}}}=1+p^{-s}+p^{-2s}+\ldots }$

${\displaystyle {\frac {\|3\mod 4,x\|-\|1\mod 4,x\|}{{\sqrt {x}}/\log x}}\approx 1+2\sum {\frac {\sin(\gamma \log x)}{\gamma }},L\left({\frac {1}{2}}+i\gamma \right)=0,L(s)={\frac {1}{1^{s}}}-{\frac {3}{1^{s}}}+{\frac {5}{1^{s}}}-{\frac {7}{1^{s}}}+\cdots }$

${\displaystyle {\vec {X}}={\begin{pmatrix}X_{1}\\X_{2}\\\cdots \\X_{n}\end{pmatrix}}}$ ${\displaystyle {\vec {S}}_{X}={\begin{pmatrix}X_{1}+X_{2}+\cdots +X_{n}\\X_{2}+\cdots +X_{n}\\\cdots \\X_{n}\end{pmatrix}}}$ ${\displaystyle {\vec {S}}_{X}={\begin{pmatrix}1&1&\cdots &1\\0&1&\cdots &1\\\cdots &\cdots &\cdots &\cdots \\0&0&\cdots &1\end{pmatrix}}{\vec {X}}}$ ${\displaystyle S_{i,j}=\sum _{k=0}^{N-\max\{i,j\}}{X_{i+k,j+k}}}$ ${\displaystyle E=E(X1_{X>x_{0}})=\mu \left(1-\Phi {\Bigl (}{\frac {x_{0}-\mu }{\sigma }}{\Bigr )}\right)+{\frac {\sigma }{\sqrt {2\pi }}}e^{-{\frac {1}{2}}\left({\frac {x_{0}-\mu }{\sigma }}\right)^{2}}}$ ${\displaystyle E^{Q}\left(e^{iuM_{T}}\right)=E^{Q}\left(e^{iu\sigma \mu _{T}+iu\sum {J_{i}}}\right)=E^{Q}\left(e^{iu\sigma \mu _{T}}\right)E^{Q}\left(e^{iu\sum {J_{i}}}\right)=e^{-{\frac {\sigma ^{2}u^{2}T}{2}}}e^{-\lambda T{\bigl (}1-\phi _{J}(u){\bigr )}},\phi _{J}(u)=E^{Q}(e^{iuJ})}$

${\displaystyle r_{t+1}=r_{t}(1-\theta )+\theta \mu +\sigma {\sqrt {r_{t}}}\delta W_{t}}$

${\displaystyle r=1-\prod _{i=1}^{N}(1-R_{i})^{\frac {1}{N}}}$

${\displaystyle {\bigl (}1+{\sqrt {2}}{\bigr )}^{1000}}$

${\displaystyle {\bigl (}1+{\sqrt {2}}{\bigr )}^{1000}+{\bigl (}1-{\sqrt {2}}{\bigr )}^{1000}}$