TITLE: When is Chebyshev's $\vartheta(x)>x$?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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 $\theta(p_{n+1})>p_{n+1}$.