Why is the series representation of the logarithm of the zeta function analytic?

Question

I try to prove, that the logarithm of the $\zeta$-function has the following representation for $z \in \mathbb{C}$ with $\text{Re}(z) > 1$ $$\log\zeta(z)=-\sum_p\log\left(1-\frac{1}{p^z}\right).$$ My idea was to show this identity for real argument and then to use the analytic continuation theorem, but I don't know, how to show that the right side is analytic. I have thought on the Weierstrass M-test and I hoped that I can try something like $$\left|\log\left(1-\frac{1}{p^z}\right)\right| \leq (1+\varepsilon)\frac{1}{p^{1+\delta}} $$ for all $z \in \mathbb{C}$ with $\text{Re}(z) \geq 1 + \delta$ for any fixed $\delta > 0$, but I failed. I will be grateful, if you could help me to show that the right side is analytic.