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

Kuok Fai Chao, Roger Plymen

Published 2005-09-14, updated 2009-03-23Version 7

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 $\pi(x)-\li(x) > 3.2 \times 10^{151}$. There are at least $10^{154}$ successive integers $x$ in this interval for which $\pi(x)>\li(x)$. This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.