I am asking this question on MO as earlier it was asked on MSE( link below) . I even put bounty on it and waited but no one answered. So, I am posting it here as I have no option and I am unable to think about it.

https://math.stackexchange.com/questions/3543021/unable-to-think-how-to-prove-a-result-in-a-research-paper-of-wadim-zudilin-and-t

Question is ->I am self studying research paper A note on odd zeta values and I am unable to think how to deduce a result which authors doesn't proves. This result has to be proved assuming prime number theorem and it's on Page12 of paper.

Prove that $\lim_ {n\to\infty} \frac{\log(\Phi_n) } {n} =\int_0^{1} \rho_0 (t) d(\psi(t) + 1/t) $, where $\psi(t) $ = $\frac {\Gamma'(t) } {\Gamma(t) } $ .

Images ->

Images are for function $\Phi(n) $ and $\rho (n) $.

Can someone please tell how to prove this result. I shall be really thankful.