A new bound for the smallest

$x$ with

$\pi \left(x\right)>\text{li}\left(x\right)$.

*(English)* Zbl 1042.11001
Summary: Let $\pi \left(x\right)$ denote the number of primes $\le x$ and let $\text{li}\left(x\right)$ denote the usual integral logarithm of $x$. We prove that there are at least ${10}^{153}$ integer values of $x$ in the vicinity of $1\xb739822\times {10}^{316}$ with $\pi \left(x\right)>\text{li}\left(x\right)$. This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of $\pi \left(x\right)-\text{li}\left(x\right)$ 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 $1\xb7617\times {10}^{9608}$, where $\pi \left(x\right)$ appears to exceed $\text{li}\left(x\right)$ by more than $\xb718{x}^{\frac{1}{2}}/logx$. The plots strongly suggest, although upper bounds derived to date for $\text{li}\left(x\right)-\pi \left(x\right)$ are not sufficient for a proof, that $\pi \left(x\right)$ exceeds $\text{li}\left(x\right)$ for at least ${10}^{311}$ integers in the vicinity of $1\xb7398\times {10}^{316}$. If it is possible to improve our bound for $\pi \left(x\right)-\text{li}\left(x\right)$ by finding a sign change before ${10}^{316}$, our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of $\text{li}\left(x\right)-\pi \left(x\right)$ and find that as $x$ departs from the region in the vicinity of $1\xb762\times {10}^{9608}$, the density is $1-2\xb77\times {10}^{-7}=\xb799999973$, 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.

##### MSC:

11-04 | Machine computation, programs (number theory) |

11A15 | Power residues, reciprocity |

11M26 | Nonreal zeros of $\zeta \left(s\right)$ and $L(s,\chi )$; Riemann and other hypotheses |

11Y11 | Primality |

11Y35 | Analytic computations |