A new bound for the smallest $x$ with $\pi (x)>li(x)$

Abstract

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann zeros, we prove that there exists $x$ in the interval $[exp(727.951858),exp(727.952178)]$ for which $π(x)-\li(x) > 3.2 \times 10^{151}$. There are at least $10^{154}$ successive integers $x$ in this interval for which $π(x)>\li(x)$. This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.

Final version, to be published in the International Journal of Number Theory [copyright World Scientific Publishing Company][www.worldscinet.com/ijnt]