I am interested in knowing whether the following statements are true for $t\in[0,\infty)$. If so, how do I prove them, or can you provide a reference?
$$\text{Re}\frac{\zeta \left(2 i t+2\right)}{\zeta (i t+1)}=O(1)$$
$$\text{Im}\frac{ \zeta \left(2 i t+2\right)}{\zeta (i t+1)}=O(1) $$
This questions are now answered as per Conrad's comments, which I copy paste here (I had to rewrite the texed parts, I hope I did not introduce any typos:
$1/10\le|\zeta(2+it)|\le 2$ while $1/|\zeta(1+it)|$ is unbounded (there is $t_n\rightarrow\infty$,$1/|\zeta(1+it_n)|>>\log \log t_n$) so the above cannot be true (technically it would follow that at least one of the relations cannot be true but I think that the proof in Titchmarsh of the unboundedness of $1/|\zeta(1+it)|$ can be adapted easily to show that both the above are unbounded) ...
$t_n$ is just a sequence that grows to infinity on which $1/|\zeta(1+it_n)|>c \log\log t_n$ for some constant; it is known that $1/\zeta(1+it)=O(\log t/\log\log t$) though on RH the expected bound is just the one above so $O(\log\log t)$; this is in Titchmarsh too; not sure though if there are estimates for $t_n$ above on which $1/|\zeta(1+it_n)|>c\log\log t_n$, just that such exists (while $c$ above can be taken $6/\pi^2−\epsilon$)