Page 199 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Nonsense! The inequalities mean that 1 > 1 which is false. So, by contradiction we have proven that

there is only one real number y such that

y
e = x,

and

log(x)= y.

∞
The Prime Number Theorem uses the natural logarithm to describe how quickly the sequence {P n } n=1
marches towards infinity.

Theorem 7.5.3 (Prime Number Theorem). For n ∈ N , define

P n =the number of primes less than or equalto n.

Then, let

P n
x n = .
n/ log(n)

∞ converges to 1,
The sequence {x n } n=1

P n
lim =1.
n→∞ n/ log(n)

What does it mean? For any ghost number > 0, there exists a giant number N ∈ N such that

∀n> N, |x n − 1| <.

So, for example, there exists N ∈ N such that

P n
− 1 < 0.1, ∀n> N.

n/ log(n)

This means that

P n
0.9 < < 1.1, ∀n> N,
n/ log(n)

which we can re-arrange to

n n
0.9 <P n < 1.1 , ∀ n> N.
log(n) log(n)

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