# Definition:Euler Product

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## Definition

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

Let $\ds \map f s = \sum_{n \mathop \in \N} a_n n^{-s}$ be its Dirichlet series.

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

From Product Form of Sum on Completely Multiplicative Function, for $\map \Re s > \sigma_a$ we have:

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

where $p$ ranges over the primes.

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

This article is incomplete.In particular: Completely multiplicative hypothesis not mentioned. Needs also the statement: $\ds \map f z = \prod_p \set {\sum_{k \mathop \ge 1} a_{p^k} p^{-k s} }$ or however it goes for multiplicative functions which are not completely multiplicativeYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Stub}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

## Source of Name

This entry was named for Leonhard Paul Euler.