When is Chebyshev's $\vartheta(x)>x$? Various bounds and computations for Chebyshev's functions
$$
\vartheta(x) = \sum_{p\le x} \log p, \quad \psi(x) = \sum_{p^a\le x} \log p
$$
can be found in e.g.

*

*Rosser and Schoenfeld, Approximate Formulas for Some Functions of Prime Numbers

*Dusart, Estimates of Some Functions Over Primes without R.H.

*Nazardonyavi and Yakubovich, Sharper estimates for Chebyshev’s functions $\vartheta$ and $\psi$
Nazardonyavi and Yakubovich cite Ingham to give as Theorem 1.14
$$
\psi(x)-x = \Omega_{\pm}\left(x^{1/2}\log\log\log x\right)
$$
and Wikipedia cites Hardy and Littlewood already with
$$
\psi(x)-x \neq o\left(x^{1/2}\log\log\log x\right)
$$
(though it doesn't say explicitly that large deviations should be on both sides, but follows the two-sided result of Schmidt). These suggest that there should be infinitely many $x$ with $\psi(x)-x>A\sqrt{x}$ for any given constant $A$.
But Dusart gives for all $x>0$
$$
\psi(x)-\vartheta(x)<1.00007\sqrt{x}+1.78\sqrt[3]{x}
$$
which is bounded by a constant multiple of $\sqrt{x}$.
Hence (if I read correctly) there should be infinitely many $x$ with $\vartheta(x)>x$. Is any such value for $x$ known, or a range where this is expected to occur?
 A: Littlewood’s oscillation theorem says that there is a positive constant $C$ such that for infinitely many numbers $x$, 
$$
\theta(x) > x + C\sqrt{x}\log \log \log x.
$$
A discussion about this can be found in the article Efficient prime counting and the Chebyshev primes. It is related to the study of $\epsilon(x)=\mathrm{Li}(x)-\pi(x)$, and $\epsilon_{\theta(x)}=\mathrm{Li}(\theta(x))-\pi(x)$. The function $\epsilon(x)$ is known to be positive up to the very large Skewes’ number. On the other hand it is known that RH is equivalent to the fact that $\epsilon_{\theta(x)}$ is always positive. Concerning $\theta$ see also Lemma $1.6$ in the above article for $\theta(p_{n+1})>p_{n+1}$.
A: The analogous problem of finding an explicit number $x$ where $\pi(x) > \mathop{\rm li}(x)$ is related to "Skewes' number". For that problem, Bays and Hudson show that such an $x$ can be found less than about $1.4\times10^{316}$; this has been modestly improved a few times since then (I think most recently by Zegowitz). Based on approximate computations, it seems reasonable to suspect that that's actually the size of the smallest such $x$.
People don't seem to have looked as much at specific $x$ for which $\theta(x)>x$, but the same techniques apply, and I suspect that the smallest such occurrence is basically the same as that of $\pi(x) > \mathop{\rm li}(x)$.
