Euler product formula for Dirichlet series of multiplicative function

Statement

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

$\sum _{n\in \mathbb {N} }{\frac {f(n)}{n^{s}}}=\prod _{p\in \mathbb {P} }\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}$.

Such an expression is sometimes termed an Euler product.

Related facts

• Euler product formula for Dirichlet series of completely multiplicative function
• Dirichlet series of Dirichlet product equals product of Dirichlet series