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Answer by GH from MO for Tight prime bounds
The inequality you propose would imply that$$ |\operatorname{li}(n)-\pi(n)| \ll \frac{\sqrt{n}}{\log n}. $$On the other hand, Littlewood (1914) showed that this relation is false, in fact he proved...
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This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the...
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