A NEW BOUND FOR THE SMALLEST x WITH π(x) > li(x)

Let π(x) denote the number of primes ≤ x and let li(x) denote the usual integral logarithm of x. We prove that there are at least 10153 integer values of x in the vicinity of 1.39822× 10316 with π(x) > li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of π(x)− 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 × 109608, where π(x) appears to exceed li(x) by more than .18x 1 2 / log x. The plots strongly suggest, although upper bounds derived to date for li(x) − π(x) are not sufficient for a proof, that π(x) exceeds li(x) for at least 10311 integers in the vicinity of 1.398 × 10316. If it is possible to improve our bound for π(x)− li(x) by finding a sign change before 10316, our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of li(x) − π(x) and find that as x departs from the region in the vicinity of 1.62 × 109608, the density is 1 − 2.7 × 10−7 = .99999973, and that it varies from this by no more than 9 × 10−8 over the next 1030000 integers. This should be compared to Rubinstein and Sarnak.