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.