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VII. Simple Analytic Proof of the Prime Number Theorem
72
where δ(N) tends to 0, and our trick is to pick a function w(N) which

approaches ∞ but such that w(N)δ N approaches 0.
w(N)
This done, we may conclude that

N N
D(n) N + O √ + O Nw(N)δ
w(N) w(N)
n≤N
N + o(N),
and the proof is complete.

Second Proof of the Prime Number Theorem.

In this section, we begin with Tchebychev’s observation that

log p
− log n is bounded, (12)
p
p≤n
which he derived in a direct elementary way from the prime
factorization on n!
The point is that the Prime Number Theorem is easily derived from

log p
− log n converges to a limit, (13)
p
p≤n
by a simple summation by parts, which we leave to the reader. Nev-
ertheless the transition from (12) to (13) is not a simple one, and we
turn to this now.
So, for z> 1, form the function

∞
1 log p log p 1
fàz) .
n z p p n z
n 1 p≤n p n≥p
Now

1 1 ∞
1 −{t}
+ z dt
n z (z − 1)p z−1 t z+1
n≥p p

p 1
+ A p (z)
z
(z − 1) p − 1