# Euler product

If $f$ is a multiplicative function, then

 $\sum_{n=1}^{\infty}f(n)=\prod_{p\text{ is prime}}(1+f(p)+f(p^{2})+\cdots)$ (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\prod_{p $\displaystyle=\sum_{k_{1}}f(p_{1}^{k_{1}})\sum_{k_{2}}f(p_{2}^{k_{2}})\cdots% \sum_{k_{t}}f(p_{t}^{k_{t}})$ $\displaystyle=\sum_{k_{1},k_{2},\ldots,k_{t}}f(p_{1}^{k_{1}})f(p_{2}^{k_{2}})% \cdots f(p_{t}^{k_{t}})$ $\displaystyle=\sum_{k_{1},k_{2},\ldots,k_{t}}f(p_{1}^{k_{1}}p_{2}^{k_{2}}% \cdots p_{t}^{k_{t}})$ $\displaystyle=\sum_{P_{+}(n)

where $p_{1},p_{2},\ldots,p_{t}$ 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

 $\left\lvert\sum_{n=1}^{\infty}f(n)-\sum_{P_{+}(n)

which tends to zero as $y\to\infty$. ∎

## Examples

• If the function $f$ is defined on prime powers by $f(p^{k})=1/p^{k}$ for all $p and $f(p^{k})=0$ for all $p\geq x$, then allows one to estimate $\prod_{p

 $\prod_{p\sum_{n\ln x.$

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

 $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}\qquad\text{for \Re s>1}.$

Since the series converges absolutely, the Euler product for the zeta function is

 $\zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}\qquad\text{for \Re s>1}.$

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 EulerProduct 2013-03-22 14:10:58 2013-03-22 14:10:58 bbukh (348) bbukh (348) 8 bbukh (348) Definition msc 11A05 msc 11A51 MultiplicativeFunction RiemannZetaFunction