New bounds in R.S. Lehman's estimates for the difference $\pi\left( x\right) -li\left( x\right) $

TL;DR

This paper improves bounds on the difference between the prime counting function and the logarithmic integral, refining Lehman's method to better identify where their sign changes occur, with numerical insights near 10^316.

Contribution

The authors enhance Lehman's bounds by refining error terms and removing size restrictions, enabling more precise numerical detection of sign change points.

Findings

Improved bounds on the sign change of π(x) - li(x).

Numerical results near 10^316 for crossover regions.

Elimination of the lower size condition for error term η.

Abstract

We denote by $\pi\left( x\right) $ the usual prime counting function and let $li\left( x\right) $ the logarithmic integral of $x$. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for $\pi\left( x\right) -li\left( x\right) $ for some value $x$ not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term $S_{3}$ in his original paper. We are now able to eliminate the lower condition for the size-length $\eta$ completely. For further numerical computations this enables us to establish sharper results on the positions for the sign changes. We illustrate with some numerical computations on the lowest known crossover regions near $10^{316}$ and we discuss numerically on potential crossover regions below this value.