Theorem (Euler product). Assuming that:

Then

D_{f} (s) = \prod_{p} (1 + \frac{f (p)}{p^{2}} + \frac{f (p^{2})}{p^{2 s}} + \dots),

for $Re (s) > 1$ . Furthermore, if $f$ is completely multiplicative, then

D_{f} (s) = \prod_{p} {(1 - \frac{f (p)}{p^{s}})}^{- 1},

for $Re (s) > 1$ .