# When is $\vartheta(x)>x$? [Skewes number analog]

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 $\vartheta$.

• 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/… – Stopple Dec 18 '14 at 17:29
• 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. – Greg Martin Dec 18 '14 at 18:54
• @GregMartin: Of course -- but I wanted a citation, thus the question. :) – 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 $\theta(x)>x$.