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A new bound for the smallest x with π(x)>li(x). (English) Zbl 1042.11001
Summary: 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 10 153 integer values of x in the vicinity of 1·39822×10 316 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 1·617×10 9608 , where π(x) appears to exceed li(x) by more than ·18x 1 2 /logx. 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 10 311 integers in the vicinity of 1·398×10 316 . If it is possible to improve our bound for π(x)-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 li(x)-π(x) and find that as x departs from the region in the vicinity of 1·62×10 9608 , 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 10 30000 integers. This should be compared to Rubinstein and Sarnak.

11-04Machine computation, programs (number theory)
11A15Power residues, reciprocity
11M26Nonreal zeros of ζ(s) and L(s,χ); Riemann and other hypotheses
11Y35Analytic computations