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

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 $[exp(727.951858), exp(727.952178)]$forwhich$\pi(x)-\li(x) > 3.2 \times 10^{151}$. There are at least ${10}^{154}$ successive integers $x$ in this interval for which $\pi (x)>\text{li}(x)$. This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.