A still sharper region where 𝜋(𝑥)-𝑙𝑖(𝑥) is positive

Saouter, Yannick; Trudgian, Timothy; Demichel, Patrick

doi:10.1090/S0025-5718-2015-02930-5

A still sharper region where $\pi (x)-{\mathrm {li}}(x)$ is positive
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by Yannick Saouter, Timothy Trudgian and Patrick Demichel PDF

Math. Comp. 84 (2015), 2433-2446 Request permission

Abstract:

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.

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