Prime Numbers

1.1 Problems and progress 11
is that for random large integers x the “probability” that x is prime is about
1/ ln x.
It is interesting to ponder how Gauss arrived at his remarkable conjecture.
Thestorygoesthathecameacrosstheconjecturenumerically,bystudyinga
table of primes. Though it is clearly evident from tables that the primes thin
out as one gets to larger numbers, locally the distribution appears to be quite
erratic.SowhatGaussdidwastocountthenumberofprimesinblocksof
length 1000. This smoothes out enough of the irregularities (at low levels) for
a “law” to appear, and the law is that near x, the “probability” of a random
integer being prime is about 1/ ln x. This then suggested to Gauss that a
reasonable estimate for π(x) might be the logarithmic-integral function.
Though Gauss’s thoughts on π(x) date from the late 1700s, he did not
publish them until decades later. Meanwhile, Legendre had independently
conjectured the PNT, but in the form
π(x) ∼
x
ln x − B
(1.4)
with B =1.08366. No matter what choice is made for the number B, wehave
x/ ln x ∼ x/(ln x − B), so the only way it makes sense to include a number
B in the result, or to use Gauss’s approximation li (x), is to consider which
option gives a better estimation. In fact, the Gauss estimate is by far the better
one. Equation (1.2) implies that |π(x) − li (x)| = O(x/ ln k x) for every k>0
(where the big-O constant depends on the choice of k). Since
li (x) = x x
+
ln x ln 2
x
+ O
x ln 3
,
x
it follows that the best numerical choice for B in (1.4) is not Legendre’s choice,
but B = 1. The estimate
x
π(x) ≈
ln x − 1
is attractive for estimations with a pocket calculator.
One can gain insight into the sharpness of the li approximation by
inspecting a table of prime counts as in Table 1.1.
For example, consider x = 10 21 . We know from a computation
of X. Gourdon (based on earlier work of M. Deléglise, J. Rivat, and
P. Zimmermann) that
while on the other hand
and
π 10 21 = 21127269486018731928,
li 10 21 ≈ 21127269486616126181.3
1021 ln(1021 ≈ 21117412262909985552.2 .
) − 1