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

Abstract:

Let $\pi (x)$ denote the number of primes $\le x$ and let $\mathrm {li}(x)$ denote the usual integral logarithm of $x$. We prove that there are at least $10^{153}$ integer values of $x$ in the vicinity of $1.39822\times 10^{316}$ with $\pi (x)>\mathrm {li}(x)$. This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of $\pi (x)-\mathrm {li}(x)$ in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of $1.617\times 10^{9608}$, where $\pi (x)$ appears to exceed $\mathrm {li}(x)$ by more than $.18x^{\frac 12}/\log x$. The plots strongly suggest, although upper bounds derived to date for $\mathrm {li}(x)-\pi (x)$ are not sufficient for a proof, that $\pi (x)$ exceeds $\mathrm {li}(x)$ for at least $10^{311}$ integers in the vicinity of $1.398\times 10^{316}$. If it is possible to improve our bound for $\pi (x)-\mathrm {li}(x)$ by finding a sign change before $10^{316}$, our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of $\mathrm {li}(x)-\pi (x)$ and find that as $x$ departs from the region in the vicinity of $1.62\times 10^{9608}$, the density is $1-2.7\times 10^{-7}=.99999973$, and that it varies from this by no more than $9\times 10^{-8}$ over the next $10^{30000}$ integers. This should be compared to Rubinstein and Sarnak.