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ϕ(n) = �
3.6 Chebychev’s Theorem on the Density of Prime Numbers 45
1≤i≤r
ϕ(p ki
i ),
which by Case 2 means
ϕ(n) = �
(pi − 1) · p
1≤i≤r
ki−1 = �
p
1≤i≤r
ki �
· 1 − 1
�
= n ·
pi
�
�
1 −
1≤i≤r
1
�
.
pi
⊓⊔
For example, we have
ϕ(7) = 6;
ϕ(15) = ϕ(3 · 5) = 2 · 4=8=15· 2 4
3 · 5 ,
ϕ(210) = ϕ(2 · 3 · 5 · 7) = 1 · 2 · 4 · 6 = 48 = 210 · 1 2 4 6
2 · 3 · 5 · 7 ,
ϕ(1000) = ϕ(23 · 53 )=1· 22 · 4 · 52 = 400 = 1000 · 1 4
2 · 5 ,
ϕ(6860) = ϕ(22 · 5 · 73 )=1· 2 · 4 · 6 · 72 = 2352 = 6860 · 1 4 6
2 · 5 · 7 .
3.6 Chebychev’s Theorem on the Density of Prime
Numbers
In this section, we develop upper and lower bounds on the density of the prime
numbers in the natural numbers. Up to now, we only know that there are
infinitely many prime numbers. We want to estimate how many primes there
are up to some bound x. Results of the type given here were first proved by
Chebychev in 1852, and so they are known as “Chebychev-type estimates”.
Definition 3.6.1. For x>1, letπ(x) denote the number of primes p ≤ x.
Table 3.5 lists some values of this function, and compares it with the
function x/(ln x − 1).
x 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 30
π(x) 1 2 2 3 3 4 4 4 4 5 5 6 6 6 8 10
x 40 50 70 100 200 500 1000 5000 10000
π(x) 12 15 19 25 46 95 168 669 1229
x/(ln x − 1) 14.9 17.2 21.5 27.7 46.5 95.9 169.3 665.1 1218.0
Table 3.5. The prime counting function π(x) for some integral values of x; values
of the function x/(ln x − 1) (rounded) in comparison
The following theorem is important, deep, and famous; it was conjectured
as early as in the 18th century by Gauss, but proved only in 1896, independently
by Hadamard and de la Vallée Poussin.