More Recent Estimates
These (enormous) upper bounds have since been reduced considerably by using large scale computer calculations of zeros of the Riemann zeta function. The first estimate for the actual value of a crossover point was given by Lehman (1966), who showed that somewhere between 1.53×101,165 and 1.65×101,165 there are more than 10500 consecutive integers x with π(x) > li(x). Without assuming the Riemann hypothesis, H. J. J. te Riele (1987) proved an upper bound of 7×10370. A better estimation was 1.39822×10316 discovered by Bays & Hudson (2000), who showed there are at least 10153 consecutive integers somewhere near this value where π(x) > li(x), and suggested that there are probably at least 10311. Chao & Plymen (2010) gave a small improvement and correction to the result of Bays and Hudson. Bays and Hudson found a few much smaller values of x where π(x) gets close to li(x); the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though computer calculations suggest they are unlikely to exist. (Saouter & Demichel 2010) find a smaller interval for a crossing, which was slightly improved by (Zegowitz 2010). The same source shows that there exists a number x violating π(x) < li(x) below . The exponent could be reduced to 727.951338611, assuming Riemann hypothesis.
Rigorously, Rosser & Schoenfeld (1962) proved that there are no crossover points below x = 108, and this lower bound was subsequently improved by Brent (1975) to 8×1010, and by Kotnik (2008) to 1014.
There is no explicit value x known for certain to have the property π(x) > li(x), though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
Wintner (1941) showed that the proportion of integers for which π(x)>li(x) is positive, and Rubinstein & Sarnak (1994) showed that this proportion is about .00000026, which is surprisingly large given how far one has to go to find the first example.
Read more about this topic: Skewes' Number
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