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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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    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.2012-10-22

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