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.