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VII. Simple Analytic Proof of the Prime Number Theorem
74
In the range N to N(1 + 1), by (14), a n ≥ a N + log(N/n) ≥
N(1+1) N(1+1)
a N − 1.So a n /è ≥ (a N − 1) 1/è , and (15) yields
N N
1 2 1 2 2
1 + 21 + 1 .à 17)
N(1+1) 1 N1/N(1 + 1)
a N 1 +
N n
Similarly in [N(1 − 1), N], a n ≤ a N + log(N/n) ≤ a N + 1/(1 −
1), so that
N N
1 1
a n
≤ a N + ,
n 1 − 1 n
N(1−1) N(1−1)
and (16) gives
2
−1 1 2 −1 1 2 1 − 21
a N ≥ − N ≥ − .
1 − 1 1 1 − 1 N1/N 1 − 1
N(1−1) n
(18)
Taken together, (17) and (18) establish that a N → 0, and so (13) is
proved.