Euler Product

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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.