A QUICK PROOF OF A THEOREM OF MERTENS' We first prove a ...

A QUICK PROOF OF A THEOREM OF MERTENS’
LEO GOLDMAKHER
We first prove a weak form of Stirling’s formula:
log n
nx
=
x
x
[t]
log t d[t] = [x] log x −
1
1 t dt
=
x
{x}
x log x − {x} log x − x + 1 +
t dt
We also know that
Λ(d) =
Λ(p j ) =
whence
d|n
p j |n
= x log x − x + O(log x)
p|n
jordp(n)
nx
1
log p =
(ordp n) log p = log n
x log x − x + O(log x) =
log n =
Λ(d) =
x
Λ(d)
d
Since ψ(x) ≪ x, we deduce that
dx
= x
Λ(d)
d
dx
nx
p|n
d|n
Λ(d)
d + O ψ(x) .
= log x + O(1).
But this sum is essentially the sum over the primes:
Λ(d)
d
dx
= log p
+
p
px
px1/2 log p
+
p2 px1/3 =
log p
+ · · ·
p3
⎛
log p
+ O ⎝
p
px
px1/2 log p
p2
1
1 − 1
p
⎞
⎠
= log p
+ O(1).
p
Thus, setting
px
R(x) =
px
log p
p
− log x
we have shown that R(x) ≪ 1. We are now in the position to prove Mertens’ estimate:
1
dx

Proposition. There exists a constant C > 0 such that
1
1
= log log x + C + O
p log x
Proof.
px
1
p =
x
2 −
1
log t
px
d (log t + R(t))
x
dt R(x)
=
+
2 t log t log x +
x
R(t)
2 t log 2 t dt
∞
1
R(t)
= log log x − log log 2 + O +
log x 2 t log 2 ∞
R(t)
dt −
t x t log 2 t dt
∞
R(t)
= log log x +
t log 2
1
dt − log log 2 + O
t log x
Department of Mathematics, University of Toronto
40 St. George Street, Room 6290
Toronto, M5S 2E4, Canada
E-mail address: leo.goldmakher@utoronto.ca
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