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.