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VII. Simple Analytic Proof of the Prime Number Theorem
residue theorem,
69
1 z
S N (z + w)N z + dz
z R 2
A

1 z
z
2πiS N (w) − S N (z + w)N + 2 dz,
−A z R
with −A as usual denoting the reflection of A through the origin.
Thus, changing z to −z, this can be written as

1 z
S N (z + w)N z + dz
z R 2
A
1 z
2πiS N (w) − S N (w − zðN −z + 2 dz. (4)
A z R
Combining (3) and (4) gives
2πi[Fàw) − S N (w)]

S N (w − zð 1 z
z
r N (z + w)N − + dz (5)
N z z R 2
A
1 z
+ Fàz + w)N z + 2 dz,
B z R
and, to estimate these integrals, we record the following (here as
usual we write z x, and we use the notation α β to mean
simply that |α|≤è β|):

1 z 2x
+ along |z| R (in particular on A), à 6)
z R 2 R 2

1 z 1 |z| 2 2
+ 1 + on the line z −δ,
z R 2 δ R 2 δ
|z|≤ R, (7)
∞ ∞
1 dè 1
r N (z + w) ≤ , à 8)
n x+1 n x+1 xN x
n N+1 N
and
N N

S N (w − zð n x−1 ≤ N x−1 + n x−1 dè
n 1 0