408 Jean-Marie De Koninck
to +∞ and such that for n = 0, 1, 2, . . .,
π(x2n+1) Li(x2n+1)
x2n2 1/
+1
log log log x2n+1
log x2n+1
,
π(x2n) Li(x2n)
−x2n2 1/
log log log x2n
log x2n
;
the smallest number x = x∗ for which π(x) Li(x) is not known; interestingly,
these past years, the size of x∗ has been gradually narrowed down:
1. in 1933, Skewes shows that, assuming the Riemann Hypothesis,
x∗
10101034
;
2. in 1955, Skewes shows that, without any hypothesis, x∗
eeee7.7
, a much
larger number;
3. in 1966, Lehman shows that x∗ 1.65 ·
101165;
4. in 1986, te Riele shows that between 6.62 · 10370 and 6.69 · 10370, there are
more than 10180 successive integers x such that π(x) Li(x);
5. in 2000, C. Carter & R.H. Hudson [30] show that π(x) Li(x) for a certain
number x close to 1.39 · 10316.
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