A still sharper region where $\pi(x)-{\mathrm{li}}(x)$ is positive
Résumé
We consider the least number $x$ for which a change of sign of $\pi(x)-\mathrm{li}(x)$ occurs. First, we consider modifications of Lehman's method that enable us to obtain better estimates of some error terms. Second, we establish a new smaller upper bound for the first $x$ for which the difference is positive. Third, we use numerical computations to improve the final result.