Definition:Euler Product

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 multiplicative You 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

• Product Form of Sum on Completely Multiplicative Function

Source of Name

This entry was named for Leonhard Paul Euler.