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.