Is there an analog of the p-series test for infinite products?

Question

What I mean:

P-series: $$\sum_{n=1}^\infty\frac{1}{n^p}$$An infinite product "P-series": $$\prod_{n=1}^\infty(1+\frac{1}{n^p})$$ For what $p\in\mathbb{R}$ does the infinite series converge? Diverge? Has this been considered before?

Answer 1

Let

$$p_n=\prod_{k=1}^n\left(1+\frac1{k^p}\right)$$ the partial product and then we have $$\ln p_n=\sum_{k=1}^n\ln\left(1+\frac1{k^p}\right)$$ and since $$\ln\left(1+\frac1{k^p}\right)\sim_\infty\frac 1{k^p}$$ then the sequence $(\ln p_n)$ is convergent if and only if $p>1$ hence $(p_n)$ is also convergent if and only if $p>1$