TITLE: what do we know about the product $\prod_{n\in \mathbb{N}}(1-\frac{x^m}{n^m})?$
QUESTION [9 upvotes]: My question is what do we know about the product $\prod_{n\in \mathbb{N}}(1-\frac{x^m}{n^m})?$ which is a slightly modified product from the eulerian product.

REPLY [2 votes]: The even case $$\prod_{n \ge 1} (1-(x/n)^{2m}) =\prod_{a=1}^m \prod_{n \ge 1} (1- e^{2i \pi a/m}(x/n)^2)= \prod_{a=1}^m \frac{\sin(\pi e^{i \pi a/ m} x)}{\pi e^{i \pi a/ m} x}$$
Otherwise
$$\prod_{n \ge 1} (1-(x/n)^m) =\lim_{N \to \infty }\prod_{a=1}^m \prod_{n=1}^N (1- e^{2i \pi a/m}x/n)= \prod_{a=1}^m \frac{-e^{2i \pi a/ m} x}{\Gamma(-e^{2i \pi a/ m} x)}$$