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This question is about Chebyshev's first function, $\vartheta(x) = \sum_{p\leq x}\log p.$

Assuming the truth of the Riemann hypothesis, $|\vartheta(x) -x|= O(x^{1/2+\epsilon})$ for $\epsilon > 0.$

See, e.g., this note.

My question is, do we have any reason not to think that (for example)

$|\vartheta(x) - x| < \sqrt{2x}~ ,$

might be true, however remote the likelihood of showing it? Thanks.

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I believe--but may be wrong--that the best known lower bound is something like $\Omega_\pm(\sqrt x/\log^kx),$ which would allow $\sqrt{2x}$. Hopefully an expert can chime in. – Charles Oct 19 '12 at 15:24
I'm not a number theorist (let alone an analytic number theorist) so I don't guarantee that it is. But your question is interesting and I have high hopes that one will stop by to resolve this issue. – Charles Oct 19 '12 at 17:57
Possibly useful: Littlewood showed in 1914 that $\pi(x)-\operatorname{Li}(x)=\Omega_\pm\left(\sqrt x\frac{\log\log\log x}{\log x}\right).$ If you can find a proof it probably gives similar bounds on $\vartheta(x)-x.$ – Charles Oct 22 '12 at 20:37
Yes, and Dusart (2010) updates that bound using modern verification of the RH up to large finite bounds. But that gives only an upper bound on variability, while the OP seems to seek a lower bound. – Charles Oct 22 '12 at 22:26

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