Page 200 - 'Blast_Into_Math

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

The definition says that for any > 0, no matter how small, we can find a giant number N ∈ N such that

n n
(1 − ) <P n < (1 + ) ∀n> N.
log(n) log(n)

This means that, the number of primes less than or equal to n is approximately

n
,
log(n)

and this “approximately” gets better and better as n gets larger and larger.

7.5.2 The Riemann zeta function

The Prime Number Theorem can be proven by many methods, but any of these methods would require
at least 50 pages of hard work. Most analytic number theorists find the proof which uses complex analysis
to be the simplest, and therefore the most mathematically beautiful. What makes analysis complex? The

imaginary number i.

Definition 7.5.4 Define the imaginary number i so that

2
i = −1.

A complex number is

z = a + i ∗ b,

where a and b are both real numbers, and a is known as the real part of z , and b is known as the
imaginary part of z . The set of all complex numbers is

C = {a + i ∗ b such that a ∈ R, and b ∈ R}.

Addition, subtraction, multiplication, and division all follow the same rules with complex numbers.
Perhaps we will go on to do complex analysis together in another book, or perhaps you will learn complex
analysis from Ahlfors [Ah].

The Prime Number Theorem was first proven by using complex analysis to understand the Riemann
zeta function. The Riemann zeta function is named after Bernhard Riemann, and written using the

Greek letter ζ , pronounced zeta. If x is a real number and x> 1, then the Riemann zeta function is

∞
−x

ζ(x)= n .
n=1

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