2025/01/08 by Michael Revers, Revers, Michael
Mathematics · #11M26 #11N05 #11Y11 #11Y35 #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2501.04488
openalex publication_date 2025/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We denote by π( x) the usual prime counting function and let li( x) 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 π( x) -li( x) 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 S3 in his original paper. We are now able to eliminate the lower condition for the size-length η 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 10316 and we discuss numerically on potential crossover regions below this value.