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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.

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    What have you done? I claim it's obvious *which* $z$ maximizes $f_k(r)$ to get an explicit formula for $M(r)$. Taylor/Laurent series are often useful; did you get anywhere with those?2012-03-17
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    @Hurkyl: I fairly new to this...can u help me start.2012-03-17

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