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