A PRIMER OF ANALYTIC NUMBER THEORY: From Pythagoras to ...

Theorem. The Prime Number Theorem is true:
�(x) ∼ Li(x) ∼ x
log(x) .
10.4 Riemann’s Formula 249
We won’t give the full proof here; you can find it in any other book on
analytic number theory. But here are some of the ideas. We saw in the previous
section that Li(x) ∼ x/ log(x). And (10.10) shows that that �(x) ∼ �(x).
The main idea of the theorem is that the first term written in Riemann’s
explicit formula for �(x), the Li(x) term, is actually the most significant. It
dominates all the others. We already mentioned the fact that this term comes
from the pole of �(s) ats = 1; that is, the Harmonic series diverges. In fact,
it is known that the Prime Number Theorem is equivalent to the following
theorem, which is an improvement on (10.2).
Theorem. The nontrivial zeros � of �(s) satisfy
0 < Re(�) < 1. (10.15)
Proof. This deep result begins with the trivial trig identity that for any
angle �,
3 + 4 cos(�) + cos(2�) = 2(1 + cos(�)) 2 ≥ 0.
We will make use of this in the series expansion for Re(log(�(s))):
� �
�
Re log (1 − p −s ) −1
�� �
� ∞� 1
= Re
k p−ks
�
p
= �
p
p
∞�
k=1
k=1
1
k p−k� cos(kt log(p)).
We apply this to �(1 + �), to �(1 + � + it), and to �(1 + � + 2it) and take the
following clever linear combination:
3Re(log(�(1 + �))) + 4Re(log(�(1 + � + it))) + Re(log(�(1 + � + 2it)))
= � ∞� 1
k p−k(1+�) (3 + 4 cos(kt log(p)) + cos(k2t log(p))) .
p
k=1
According to the trig identity, this mess is ≥ 0. But the facts about logarithms
in Interlude 3 imply that the left side is actually
3 log(|�(1 + �)|) + 4 log(|�(1 + � + it)|) + log(|�(1 + � + 2it)|).