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).

  • 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 '19 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 '19 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. $\endgroup$ Mar 23 '19 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 '19 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 '19 at 14:48

$\vert\psi(x)-x\vert \leq \sqrt{x}\log^2x/(8\pi)$ is equivalent to RH. See Theorem 4.9 of the book "Equivalents of the Riemann Hypothesis" by Broughan (2017).

  • $\begingroup$ Thanks for the reference. $\endgroup$
    – EGME
    Apr 27 '19 at 9:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.