Large t closed-form asymptotics for Riemann $\Xi(t)$ function or $Z(t)$ function

Question

Is there a large $t$ closed-form asymptotics for Riemann $\Xi(t)$ function or Riemann-Siegel $Z(t)$ function? For example $$\Xi(t)=A(t)\cos(\tfrac{t}{2}\log(\tfrac{t}{\pi})+b).$$

Answer 1

No such result is known. A true, understandable asymptotic for $Z(t)$ is likely of comparable strength to the Riemann Hypothesis. It would certainly be of comparable strength to the Lindelof Hypothesis (which is weaker than RH, but closely related).

The Lindelof Hypothesis states that $\lvert \zeta(1/2 + it) \rvert = \lvert Z(t) \rvert \ll \lvert t \rvert^\epsilon$ for all sufficiently large $t$, for any $\epsilon > 0$. This is a statement purely about growth, but is completely unknown. The best that we know right now is that $\lvert Z(t) \rvert \ll \lvert t \rvert^{.155}$ or so, due to Bourgain.