Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$

Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing $\pi(x)>\operatorname{li}(x)$, and Kotnik gives a lower bound, but I don't know what has been proved on the more fundamental (?) question of $\vartheta$.