1. PANORAMA OF ARITHMETIC FUNCTIONS 5
Now we are going to give some basic examples of arithmetic functions. We
begin by exploiting Dirichlet convolutions. The first one is the divisor function
τ = 1 1, τ(n) =
d|n
1,
ζ2(s)
=

1
τ(n)n−s.
This is multiplicative with
τ(pα)
= α + 1. We have
ζ4(s)
ζ(2s)
=

1
τ(n)2n−s
ζ3(s)
ζ(2s)
=

1
τ(n2)n−s
ζ3(s)
=

1 d2m=n
τ(m2) n−s.
Note that
(1.13)
d2m=n
τ(m2)
=
klm=n
1 = τ3(n),
say. Next we get
ζ2(s)
ζ(2s)
=

1
2ω(n)n−s
where ω(n) denotes the number of distinct prime divisors of n, so
2ω(n)
is the number
of squarefree divisors of n. The characteristic function of squarefree numbers is
μ(n) =
μ2(n)
=
d2|n
μ(d),
its Dirichlet series is
ζ(s)
ζ(2s)
=

1
|μ(n)|n−s
=
p
1 +
1
ps
.
Inverting this we get the generating series for the Liouville function λ(n)
ζ(2s)
ζ(s)
=

1
λ(n)n−s.
Note that
λ(n) =
(−1)Ω(n)
where Ω(n) is the total number of prime divisors of n (counted with the multiplic-
ity).
The Euler ϕ-function is defined by ϕ(n) =
|(Z/nZ)∗|;
it is the number of
reduced residue classes modulo n. This function satisfies
ϕ(n) = n
p
1
1
p
= n
d|n
μ(d)
d
.
Hence
ζ(s 1)
ζ(s)
=

1
ϕ(n)n−s.
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