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 $\vartheta$.



When is $\vartheta(x)>x$? [Skewes number analog]
Platt and Trudgian show in <http://arxiv.org/abs/1407.1914> that
$$
\theta(x)<x\quad\text{for}\quad x<1.39\cdot 10^{17}
$$
and there is an $x<\exp(727.951332668)<1.4\cdot 10^{316}$ for which $\theta(x)>x$.


