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 $\vartheta$.

  • 2
    $\begingroup$ Schoenfeld said "one can show $\theta(x)<x$ for $x<10^{11}$" in ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/… $\endgroup$ – Stopple Dec 18 '14 at 17:29
  • 1
    $\begingroup$ I'm pretty sure that all of the techniques giving results on when $\pi(x)>li(x)$ (for example, Bayes/Hudson) could be easily converted to give results on when $\theta(x)>x$, and the results would be nearly identical. $\endgroup$ – Greg Martin Dec 18 '14 at 18:54
  • $\begingroup$ @GregMartin: Of course -- but I wanted a citation, thus the question. :) $\endgroup$ – Charles Dec 18 '14 at 21:37

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 $\theta(x)>x$.


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.