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

Saouter, Yannick; Demichel, Patrick

doi:10.1090/S0025-5718-10-02351-3

A sharp region where $\pi (x)-{\mathrm {li}}(x)$ is positive
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by Yannick Saouter and Patrick Demichel PDF

Math. Comp. 79 (2010), 2395-2405 Request permission

Abstract:

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.

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