PRODUCTS OVER PRIME NUMBERS [2/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\left(1+\dfrac{1}{p(p+1)}\right)=\prod_{p\in\mathbb{P}}\dfrac{1-p^{-3}}{1-p^{-2}}=\dfrac{\zeta(2)}{\zeta(3)}=\dfrac{\pi^2}{6\zeta(3)}\]
\[\displaystyle\prod_{p\in\mathbb{P}}\left(1+\dfrac{1}{p(p-1)}\right)=\prod_{p\in\mathbb{P}}\dfrac{1-p^{-6}}{(1-p^{-2})(1-p^{-3})}=\dfrac{\zeta(2)\zeta(3)}{\zeta(6)}=\dfrac{315}{2\pi^4}\zeta(3)\]
#PrimeProducts #RiemannZetaFunction #EulerProduct #ZetaFunction #InfiniteProduct #NumberTheory

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s+1}{p^s-1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s-1}{p^s+1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct

SOME PRODUCTS OVER PRIME NUMBERS [1/2]:
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}=\prod_{p\in\mathbb{P}}\left(\sum_{k=0}^\infty\dfrac{1}{p^{ks}}\right)=\sum_{n=1}^\infty\dfrac{1}{n^s}=\zeta(s)\]
\[\displaystyle\prod_{p\in\mathbb{P}}\dfrac{p^s+1}{p^s+1}=\prod_{p\in\mathbb{P}}\left(1+\frac{2}{p^s}+\frac{2}{p^{2s}}+\cdots\right)=\sum_{n=1}^\infty\dfrac{2^{\omega(n)}}{n^s}=\dfrac{\zeta(s)^2}{\zeta(2s)}\]
#PrimeProducts #EulerProduct #RiemannZetaFunction #ZetaFunction #InfiniteProduct