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First Proof of the Prime Number Theorem.
The PNT is equivalent to the fact that the average of the coefficients
of ζ (z) is equal to 1. For simply note that 71
ζ
ζ d d 1
− (z) − log ζ(z) − log
ζ dz dz 1 − p −z
p
d p −z log p
−z
log 1 − p
dz 1 − p −z
p p
log p
.
p −z − 1
p
D(n)
This last series is the same as z where D(n) is log p whenever
n
n is a power of p, p any prime, and 0 otherwise. So indeed the average
of these coefficients is 1 D(n) whose limit being 1 is exactly
N n≤N
the Prime Number Theorem.
ζ
In short, we want the average value of the coefficients of − (z)−
ζ
ζ(z) to approach 0. Writing this function as

1 µ(n) log n d(n)

(z)[−ζ (z) − ζ(z)] − ,
ζ n z n z n z
we may write this average (of the first N terms)as

1
µ(a)[log b − dàb) ]
N
ab≤N
1 2γ
µ(a)[log b − dàb) + 2γ ] − ,
N N
ab≤N
where 2γ is chosen as the constant for which
K

[log b − dàb) + 2γ ]
b 1
√
becomes O( K).

Now we use the Landau corollary that µ(n) o(N) to
n≤N
conclude that
1
µ(n) δ(N),
N
n≤N