A sharp region where $ \pi(x)-{\mathrm{li}}(x)$ is positive
Résumé
In this article, we study the problem of changes of sign of $ \pi(x)-{\mathrm{li}}(x)$. We provide three improvements. First, we give better esimates of error term for Lehman's theorem. Second, we rigorously prove the positivity of this difference for a region formerly conjectured by Patrick Demichel. Third, we improve the estimates for regions of positivity by using number theoretic results.