# Euler Product

## Theorem

Let $a_n : \N \to \C$ be an arithmetic function.

Let $\displaystyle f \left({s}\right) = \sum_{n \mathop \in \N} a_n n^{-s}$ be its Dirichlet series.

Let $\sigma_a$ be its abscissa of absolute convergence.

Then for $\Re \left({s}\right) > \sigma_a$:

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

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.

## Source of Name

This entry was named for Leonhard Paul Euler.