Euler product
If $f$ is a multiplicative function^{}, then
$$\sum _{n=1}^{\mathrm{\infty}}f(n)=\prod _{p\text{is prime}}(1+f(p)+f({p}^{2})+\mathrm{\cdots})$$  (1) 
provided the sum on the left converges absolutely. The product on the right is called the Euler product^{} for the sum on the left.
Proof of (1).
Expand partial products on right of (1) to obtain by fundamental theorem of arithmetic^{}
$$  $={\displaystyle \sum _{{k}_{1}}}f({p}_{1}^{{k}_{1}}){\displaystyle \sum _{{k}_{2}}}f({p}_{2}^{{k}_{2}})\mathrm{\cdots}{\displaystyle \sum _{{k}_{t}}}f({p}_{t}^{{k}_{t}})$  
$={\displaystyle \sum _{{k}_{1},{k}_{2},\mathrm{\dots},{k}_{t}}}f({p}_{1}^{{k}_{1}})f({p}_{2}^{{k}_{2}})\mathrm{\cdots}f({p}_{t}^{{k}_{t}})$  
$={\displaystyle \sum _{{k}_{1},{k}_{2},\mathrm{\dots},{k}_{t}}}f({p}_{1}^{{k}_{1}}{p}_{2}^{{k}_{2}}\mathrm{\cdots}{p}_{t}^{{k}_{t}})$  
$$ 
where ${p}_{1},{p}_{2},\mathrm{\dots},{p}_{t}$ are all the primes between $1$ and $y$, and ${P}_{+}(n)$ denotes the largest prime factor of $n$. Since every natural number less than $y$ has no factors exceeding $y$ we have that
$$ 
which tends to zero as $y\to \mathrm{\infty}$. ∎
Examples

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If the function^{} $f$ is defined on prime powers by $f({p}^{k})=1/{p}^{k}$ for all $$ and $f({p}^{k})=0$ for all $p\ge x$, then allows one to estimate $$
$$ One of the consequences of this formula is that there are infinitely many primes.

•
The Riemann zeta function^{} is defined by the means of the series
$$\zeta (s)=\sum _{n=1}^{\mathrm{\infty}}{n}^{s}\mathit{\hspace{1em}\hspace{1em}}\text{for}\mathrm{\Re}s1.$$ Since the series converges absolutely, the Euler product for the zeta function^{} is
$$\zeta (s)=\prod _{p}\frac{1}{1{p}^{s}}\mathit{\hspace{1em}\hspace{1em}}\text{for}\mathrm{\Re}s1.$$ If we set $s=2$, then on the one hand $\zeta (s)={\sum}_{n}1/{n}^{2}$ is ${\pi}^{2}/6$ (proof is here (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2)), an irrational number, and on the other hand $\zeta (2)$ is a product of rational functions of primes. This yields yet another proof of infinitude of primes.
Title  Euler product 

Canonical name  EulerProduct 
Date of creation  20130322 14:10:58 
Last modified on  20130322 14:10:58 
Owner  bbukh (348) 
Last modified by  bbukh (348) 
Numerical id  8 
Author  bbukh (348) 
Entry type  Definition 
Classification  msc 11A05 
Classification  msc 11A51 
Related topic  MultiplicativeFunction 
Related topic  RiemannZetaFunction 