A sharp region where π(x)-li(x) is positive

In this article, we study the problem of changes of sign of π(x)− 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. 1. Previous work The problem of estimating the number of prime numbers goes back to Gauss. The function counting prime numbers is classically denoted π, i.e. π(x) = ∑ p≤x 1. In 1791, he conjectured that π(x) x log x . This result was proven in 1896 by Hadamard and de la Vallée-Poussin. In 1849, Gauss suggested that the log-integral function, defined by li(x) = limε→0 {∫ 1−ε 0 dt log t + ∫ x 1+ε dt log t } , should give a better approximation for π. Gauss also noted that the inequality π(x) < li(x) holds for any x in the interval [2, 3000000]. Since then, this property has been checked numerically up to 10 [1]. On the other hand, Littlewood proved in 1914 that the property π(x) > li(x) holds infinitely often, but he did not give any explicit value for such an x. In 1933, Skewes proved, assuming the Riemann hypothesis, that the latter inequality occurs at least once for a value x < 1