Please how to find order of $$ f_k(z) = \prod\limits_{n=1}^{\infty} \left(1-\frac{z}{n^k}\right) .$$
Let $M(r) = \max \{|f_k(z)|:|z| = r\}.$ Then order of $f_k(z)$ is defined as : $$\lambda = \limsup_{r\to \infty} \frac{\log \log M(r)}{\log r}.$$
Can someone help to solve this. Thank you.