Euler Product
Jump to navigation
Jump to search
This article is incomplete, as follows:
Completely multiplicative hypothesis not mentioned. Needs also the statement: $\displaystyle f \left({s}\right) = \prod_p \left\{{\sum_{k \mathop \ge 1} a_{p^k} p^{-k s}}\right\}$ or however it goes for multiplicative functions which are not completely multiplicative
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
 It has been suggested that this article or section be renamed.
|
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.
Completely multiplicative hypothesis not mentioned. Needs also the statement: $\displaystyle f \left({s}\right) = \prod_p \left\{{\sum_{k \mathop \ge 1} a_{p^k} p^{-k s}}\right\}$ or however it goes for multiplicative functions which are not completely multiplicative
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
When this page/section has been completed, {{Stub}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Proof
This is immediate from Product Form of Sum on Completely Multiplicative Function.
Source of Name
This entry was named for Leonhard Paul Euler.
