Dirichlet series generating function

$$G_1( \{a_n\}; s)=\sum_{n=1}^\infty \frac{a_n} {n^s}$$

PROPOSITION 01.5 (Euler product expression)   Assume $\{a_n\}$ is multiplicative in the sense that

$$\gcd(n,m)=1 \implies a_n a_m =a_{nm}$$

holds, Then we have

$$\sum_{n=1}^\infty \frac{a_n} {n^s} =\prod_{p; \text{prime}} \left( \sum_{k=0}^\infty \frac{a_{p^k} }{p^{ks}} \right).$$