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.