Consider the infinite product
$$P(x,p) = \prod_{n = 1}^\infty (1 - \dfrac{x}{n^p})$$
where $x \geq 0$ and $p \geq 0$.
Does it have a closed form?
Are there general values for $x, p$ such that this infinite product goes to $0$, $\infty$ or some finite number?
We have the general formula for integer $p\geq 2$ (Prudnikov et al. 1986, p. 754)
$$\prod_{k=1}^\infty \left(1-\frac{z^p}{k^p}\right)=-\frac{1}{z^p}\prod_{k=0}^{p-1} \frac{1}{\Gamma(-e^{2\pi ik/p}z)}.$$
Yours is the above evaluated at $z=x^{1/p}$