Prime Numbers

54 Chapter 1 PRIMES!
where p runs over primes. Conclude that there are infinitely many primes.
One possible route is to establish the following intermediate steps:
(1) Show that ⌊x⌋ n=1 1
n > ln x.
(2) Show that 1
1
n = p≤x (1 −
numbers n not divisible by any prime exceeding x.
p )−1 , where the sum is over the natural
1.21. Use the multinomial (generalization of the binomial) theorem to show
that for any positive integer u and any real number x>0,
⎛
1
⎝
u!
⎞u
1
⎠ ≤
p
1
n ,
p≤x
n≤x u
where p runs over primes and n runs over natural numbers. Using this
inequality with u = ⌊ln ln x⌋, show that for x ≥ 3,
p≤x
1
≤ ln ln x + O(lnlnlnx).
p
1.22. By considering the highest power of a given prime that divides a given
factorial, prove that
N! = ∞ p k=1 ⌊N/pk⌋ ,
p≤N
where the product runs over primes p. Then use the inequality
N N
N! >
e
(which follows from e N = ∞
k=0 N k /k! >N N /N !), to prove that
p≤N
ln p
> ln N − 1.
p − 1
Conclude that there are infinitely many primes.
1.23. Use the Stirling asymptotic formula
N N √2πN
N! ∼
e
and the method of Exercise 1.22 to show that
ln p
=lnN + O(1).
p
p≤N
Deduce that the prime-counting function π(x) satisfies π(x) =O(x/ ln x) and
that if π(x) ∼ cx/ ln x for some number c, thenc =1.