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.



Also see


Source of Name

This entry was named for Leonhard Paul Euler.