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

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.