Definition:Euler Product

Theorem
Let $\displaystyle f \left({s}\right) = \sum_{n \mathop \in \N} a_n n^{-s}$ be a Dirichlet series, absolutely convergent on $\Re \left({s}\right) > \sigma_a$ (see Abscissa of Absolute Convergence).

Then:
 * $\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-s} = \prod_p \frac 1 {1 - a_p p^{-s} }$

for all $s$ with $\Re \left({s}\right) > \sigma_a$, where $p$ ranges over the primes.

This representation for $f$ is called an Euler product for the Dirichlet series.

Proof
This is immediate from Product Form of Sum on Completely Multiplicative Function.