What are the best currently known upper bounds for $\theta(x)x$ assuming the Riemann Hypothesis, where $\theta(x)$ is the Chebyshev theta, and can someone provide the reference for this (not Wikipidia, where it states bounds of $O(x^{1/2+\epsilon})$, for any $\epsilon>0$ without a reference).
1

4$\begingroup$ The Wikipedia article on RH has a precise bound for $\psi(x)x$. Relation between $\psi(x)$ and $\theta(x)$ is elementary, see e.g. here and references there. $\endgroup$ – Wojowu Mar 23 at 9:20

$\begingroup$ There is also a result in Schoenfeld's "Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$ II", but is that currently the best ...? $\endgroup$ – EGME Mar 23 at 9:47

$\begingroup$ I know that Pierre Dusart has some results going past that paper. You might look at his papers, and then look at those who cite him to see if he still has the record. $\endgroup$ – Pace Nielsen Mar 23 at 14:43

$\begingroup$ Do you need explicit bounds, that is, with an explicit constant? Or do you want what is asymptotically best? $\endgroup$ – Nell Mar 23 at 14:46

$\begingroup$ I am curious about both. The Schoenfeld bound works well for what I am doing, but I was wondering whether there is something better, as you can then improve your results ... thanks all for comments $\endgroup$ – EGME Mar 23 at 14:48
5
$\begingroup$
$\endgroup$
$\vert\psi(x)x\vert \leq \sqrt{x}\log^2x/(8\pi)$ is equivalent to RH. See Theorem 4.9 of the book "Equivalents of the Riemann Hypothesis" by Broughan (2017).