Mean values of ratios of the Riemann zeta function

It is proved that $${\int }_T^{2T}{\mid \frac{\zeta (\frac{1}{2}+\mathrm{i}t)}{\zeta (1+2\mathrm{i}t)}\mid }^2\mathrm{d}t=\frac{1}{\zeta (2)}T\log T+(\frac{\log \frac{2}{\pi }+2\gamma -1}{\zeta (2)}-4\frac{{\zeta }^{\mathrm{\prime }}(2)}{{\zeta }^2(2)})T+O\left(T{(\log T)}^{-2023}\right),\mathrm{\forall }T⩾100.$$ For given $a\in \mathbb{N}$, we also establish similar formulas for second moments of $\mathrm{\mid }\zeta (\frac{1}{2}+\mathrm{i}t)\mathrm{/}\zeta (1+\mathrm{i}at)\mathrm{\mid }\mathrm{.}$ We have \begin{align*} \lim_{a \to \infty} \lim_{T \to \infty}\frac{1}{T \log T} \int_{T}^{2T} \left|\frac{ζ\left(\frac{1}{2}+{\rm i} t\right)}{ζ\left(1+{\rm i} at\right)}\right|^2 {\rm d} t = \frac{ζ(2)}{ζ(4)}. \end{align*}