20 1. Arithmetic Functions
1 for n = k and is zero otherwise. The calculation
n≤x
(e1(n) − 2e2(n))T
x
n
=
n≤x
(e1(n) − 2e2(n))
m≤x/n
ψ
x/n
m
=
k≤x
ψ
x
k
d|k
(e1(d) − 2e2(d))
=
k≤x
(−1)k−1ψ
x
k
may be taken as the framework of the proof of Proposition 1.1. It can be
generalized by replacing e1−2e2 with a more complicated linear combination
f of e1, e2,..., eN . Then 1 ∗ f is required to take only the values 0, ±1,
and the nonzero values of this function should start with (1 ∗ f)(1) = 1 and
alternate. Chebyshev chose the linear combination f = e1 −e2 −e3 −e5 +e30
to obtain his better estimates.
We exhibit an example of the use of M¨ obius inversion to establish an
arithmetically significant estimate. We find a quite precise bound for the
error term in an asymptotic estimate for the summatory function
Φ(x)
def
=
n≤x
φ(n)
of the Euler totient. The convolution identity of Gauss yields
n≤x
Φ
x
n
=
n≤x
d≤x/n
φ(d) =
nd≤x
φ(d) =
m≤x
d|m
φ(d)
=
m≤x
(1 ∗ φ)(m) =
m≤x
m =
1
2
[x]([x] + 1),
and then
Φ(x) =
n≤x
μ(n)
1
2
x
n
x
n
+ 1 =
1
2
n≤x
μ(n)
x
n
+ O(1)
x
n
+ O(1)
=
x2
2
n≤x
μ(n)
n2
+
O⎝x
⎛
n≤x
1
n
⎞
⎠
=
x2
2
∞
n=1
μ(n)
n2
+ O
x2
nx
1
n2
+ O(x log(x)) =
x2
2
∞
n=1
μ(n)
n2
+ O(x log(x))
by the second M¨ obius inversion formula. Absolutely convergent series may
be multiplied together to yield absolutely convergent series, and so
∞
m=1
1
m2
∞
n=1
μ(n)
n2
=
∞
N=1
1
N 2
mn=N
1·μ(n) =
∞
N=1
e(N)
N 2
= 1.
