Euler product formula for Dirichlet series of completely multiplicative function

Statement

For formal Dirichlet series

Suppose $f$ is a completely multiplicative function. Then, the following identity holds formally for the Dirichlet series of $f$:

$\sum _{n\in \mathbb {N} }{\frac {f(n)}{n^{s}}}=\prod _{p\in \mathbb {P} }{\frac {1}{1-f(p)p^{-s}}}$.

Here, $\mathbb {P}$ denotes the set of primes.

Such an expression is sometimes termed an Euler product.

Related facts

• Product formula for Dirichlet series of multiplicative function
• Dirichlet series of Dirichlet product equals product of Dirichlet series