Prove that $\prod _{k=1}^{\infty } \prod _{n=1}^{\infty } n^{\frac{k}{n^{\text{ks}} \zeta (k s)}}=e^{-\sum_{m=1}^{\infty} \sigma_1(m) \: P'(m \: s)}$

Question

Where $ \sigma_1(m) $ and $ P'(m \: s) $ are the Divisor function and the derivative of the Prime Zeta function, respectively.

I've tried to expand the left side via the Euler Product of the Zeta function and then take the logarithmic derivative, but I get hung up on the reciprocal Zeta part.

Any ideas?