# Euler product

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##### 1: 27.4 Euler Products and Dirichlet Series
###### §27.4 EulerProducts and Dirichlet Series
The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … Euler products are used to find series that generate many functions of multiplicative number theory. …
##### 2: 27.6 Divisor Sums
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
##### 3: 5.8 Infinite Products
5.8.3 $\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+\mathrm{i}y\right)}\right|^{2}% =\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$.
5.8.5 $\prod_{k=0}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k% )\cdots(b_{m}+k)}=\frac{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\cdots% \Gamma\left(b_{m}\right)}{\Gamma\left(a_{1}\right)\Gamma\left(a_{2}\right)% \cdots\Gamma\left(a_{m}\right)},$
##### 4: 16.18 Special Cases
16.18.1 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).$
##### 5: 27.3 Multiplicative Properties
27.3.3 $\phi\left(n\right)=n\prod_{p\mathbin{|}n}(1-p^{-1}),$
##### 6: 18.25 Wilson Class: Definitions
18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$
18.25.5 $h_{n}=\frac{n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right)}{(2n-1+% \sum_{j}a_{j})\Gamma\left(n-1+\sum_{j}a_{j}\right)}.$
18.25.7 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$
18.25.8 $h_{n}=n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right).$
##### 7: 27.14 Unrestricted Partitions
Euler introduced the reciprocal of the infinite product
27.14.2 $\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})=\left(x;x\right)_{% \infty},$ $|x|<1$,
##### 8: 16.11 Asymptotic Expansions
16.11.2 $H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.$
16.11.6 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),$ $|\operatorname{ph}\left(-z\right)|\leq\pi$;
16.11.7 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).$
16.11.8 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),$
16.11.9 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),$
##### 9: 5.14 Multidimensional Integrals
5.14.2 $\int_{V_{n}}\left(1-\sum_{k=1}^{n}t_{k}\right)^{z_{n+1}-1}\prod_{k=1}^{n}t_{k}% ^{z_{k}-1}\,\mathrm{d}t_{k}=\frac{\Gamma\left(z_{1}\right)\Gamma\left(z_{2}% \right)\cdots\Gamma\left(z_{n+1}\right)}{\Gamma\left(z_{1}+z_{2}+\dots+z_{n+1}% \right)}.$
5.14.4 $\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\,\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},$
5.14.5 $\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\,\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c% )\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},$
5.14.6 $\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}|\Delta(t_{1},\dots,t_{n})|^{% 2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}\right)\,\mathrm{d}t_{k}=% \frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},$ $\Re c>-1/n$.
5.14.7 $\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j $\Re b>-1/n$.
##### 10: 16.17 Definition
16.17.1 ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)={G^{m,n}_{p,q}}\left(z;{a_% {1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac{1}{2\pi\mathrm{i}}\int_{L% }\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{m}\Gamma\left(b_{\ell}-s\right% )\prod\limits_{\ell=1}^{n}\Gamma\left(1-a_{\ell}+s\right)}{\left(\prod\limits_% {\ell=m}^{q-1}\Gamma\left(1-b_{\ell+1}+s\right)\prod\limits_{\ell=n}^{p-1}% \Gamma\left(a_{\ell+1}-s\right)\right)}}\right)z^{s}\,\mathrm{d}s,$
16.17.3 $A_{p,q,k}^{m,n}(z)=\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq k\end{subarray}}^{m}\Gamma\left(b_{\ell}-b_{k}\right)\prod\limits_{% \ell=1}^{n}\Gamma\left(1+b_{k}-a_{\ell}\right)z^{b_{k}}}{\left(\prod\limits_{% \ell=m}^{q-1}\Gamma\left(1+b_{k}-b_{\ell+1}\right)\prod\limits_{\ell=n}^{p-1}% \Gamma\left(a_{\ell+1}-b_{k}\right)\right)}.$