For $\Re s>1$, I define the following functions $I(s)$ and $J(s)$, $$I(s)=\int_0^\infty\sum_{n=1}^\infty\mu(n)e^{-n^2\pi x}x^{s/2}\frac{dx}{x},$$ where $\mu(n)$ is the Möbius function, and the well known integral $$J(s)=\int_1^\infty\psi(x)\left(x^{-s/2}-x^{(1-s)/2}\right)\frac{dx}{x},$$ where $\psi(x)=\sum_{n=1}^\infty e^{-n^2\pi x}$.
Question. I've deduced the following claims i) and ii) involving the Riemann's Zeta function, and I would like to know rigurous proofs of one of those (isn't required a proof of both statements, but I would like to know if both are correct):
i) for $\Re s>1$ $$\frac{1}{\zeta(s)}\left(J(s)-\frac{1}{s(1-s)}\right)=I(s)\zeta(s),$$
and ii) also for $\Re s>1$, $$(s(1-s)J(s)-1)\left[\frac{I'(s)}{I(s)}+2\frac{\zeta'(s)}{\zeta(s)}\right]=s(1-s)J'(s)+\frac{1-2s}{s(1-s)}.$$ Thank you for reading my question.
As remark I say that then $I(s)$ and $J(s)$ are complementary factors of the Riemann Zeta function $\zeta(s)$ for $\Re s>1$, just as a terminology.