Does the series $\displaystyle\sum_{j=1}^\infty -\log (1-p_j^{-3/4})$ diverge, where $\{p_j\}$ is the set of primes in increasing order?

Question

Here, $\log$ is the natual logarithm. Is there a simple convergence test I can use? Thanks.

Answer 1

$$\sum_{j\geq1}-\log\left(1-\frac{1}{p_{j}^{3/4}}\right)=\log\left(\prod_{j\geq1}\left(1-\frac{1}{p_{j}^{3/4}}\right)^{-1}\right)\geq\log\left(\prod_{j\geq1}\left(1-\frac{1}{p_{j}}\right)^{-1}\right) $$ and the last term diverges by Mertens third theorem or using the Euler product of zeta function for $\textrm{Re}\left(s\right)>1 $ $$\zeta\left(s\right)=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}\underset{s\rightarrow1^{+}}{\longrightarrow}\infty. $$