How good is
Gauss's guess? We can see the difference by plotting
a graph.
The staircase of the primes versus Gauss's guess
Li(N)
N
Number of primes
(N) from 1 up to N.
How far Gauss's guess Li(N)
overestimates the number of primes less than
N: Li(N)-
(N)
Percentage error:Li(N)- (N)/
(N)x100%
100
25
5
20
1,000
168
10
5.95
10,000
1,229
17
1.38
100,000
9,592
38
0.396
1,000,000
78,498
130
0.166
107
664,579
339
0.051
108
5,761,455
754
0.0131
109
50,847,534
1,701
0.00335
1010
455,052,511
3,104
0.000682
1011
4,118,054,813
11,588
0.000281
1012
37,607,912,018
38,263
0.000102
1013
346,065,536,839
108,971
0.0000299
1014
3,204,941,750,802
314,890
0.00000983
1015
29,844,570,422,669
1,052,619
0.00000353
1016
279,238,341,033,925
3,214,632
0.00000115
Gauss's guess is not spot on. But how good is
it as we count higher? The best measure of how
good it is doing is to record the percentage error:
look at the difference between Gauss's prediction
for the number of primes and the true number of
primes as a percentage of the true number of primes.
We can see the percentage error getting smaller
in the graphs because as the scale gets bigger
the graphics look closer together.
Gauss's Prime Number Conjecture: The percentage
error get's smaller and smaller the further you
count.
There is a lot of evidence here but how can we
be sure that nothing really weird happens for
very large N.
In 1896 de la Vallée-Poussin, a Belgian,
and Hadamard, a Frenchman, proved that Gauss was
correct.
But here is a warning that it is far from obvious
that patterns will persist. Gauss also thought
that his guess would always overestimate the number
of primes. The evidence from the tables looks
overwhelming. But in 1912 Littlewood in Cambridge
proved Gauss was wrong. However the first time
that Gauss's guess underestimates the primes is
for N bigger than the number of atoms in the observable
universe, not a fact that experiment will ever
reveal.
(N) from 1 up to N.