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VII. Simple Analytic Proof of the Prime Number Theorem
70
N x 1 + 1 . à 9)
N x
By (6), (8), (9), on A,

S N (w − zð 1 z
z
r N (z + w)N − +
N z z R 2

1 1 1 2x 4 2
+ + ≤ + ,
x x N R 2 R 2 RN
and so, by the “maximum times length” estimate (M–L formula) for
integrals, we obtain

S N (w − zð 1 z 4π 2π
z
r N (z + w)N − z + 2 dz + .
A N z R R N
(10)
Next, by (2), (6), and (7), we obtain
1 z
Fàz + w)N z + dz
B z R 2
R 2 0 2|x| 3

M · N −δ dy + 2M n x dx (11)
δ R 2 2
−R −δ
4MR 6M
≤ + .
2
2
δN δ R log N
Inserting the estimates (10) and (11) into (5) gives
2 1 MR M
Fàw) − S N (w) + + + ,
2
R N δN δ R log N
2
and, if we fix R 3/1, we note that this right-hand side is <1 for
all large N. We have verified the very definition of convergence!

First Proof of the Prime Number Theorem.

µ(n)
Following Landau, we will show that the convergence of
n n
(as given above) implies the PNT. Indeed all we need about this

convergent series is the simple corollary that µ(n) o(N).
n≤N
Expressing everything in terms of the ζ-function, then, we have
established the fact that 1 has coefficients which go to 0 on average.
ζ(z)