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Second Proof of the Prime Number Theorem.
where A p (z) is analytic for z> 0 and is bounded by
1 |zàz − 1)| 73
+ .
x
x
p (p − 1) xp x+1
Hence,
 
1 log p
fàz)  + A(z) ,

z
z − 1 p − 1
p
where A(z) is analytic for z> 1 by the Weierstrass M-test.
2
By Euler’s factorization formula, however, we recognize that
log p −d
log ζ(z),
z
p − 1 dz
p
and so we deduce, by (1), that fàz) is analytic in z ≥ 1 except for
2
a double pole with principal part 1/(z − 1) + c/(z − 1) at z 1.
Thus if we set
a n

Fàz) fàz) + ζ (z) − cζ(z)
n z
n
where
log p
a n − log n − c, (14)
p
p≤n
we deduce that Fàz) is analytic in z ≥ 1.
From (12) and our convergence theorem, then, we conclude that
a n

converges,
n
and from this and the fact, from (14), that a n +log n is nondecreasing,
we proceed to prove a n → 0.
By applying the Cauchy criterion we find that, for N large,
N(1+1)

a n 2
≤ 1 (15)
n
N
and
N
a n 2

≥ð 1 . à 16)
n
N(1−1)