1.5. Sums over divisors 17

operation on arithmetic functions is the Dirichlet convolution. If f and g

are arithmetic functions then

(f ∗ g)(n) =

d|n

f(d)g

n

d

=

km=n

f(k)g(m)

is their Dirichlet convolution. It is a straightforward exercise to see that

under addition and Dirichlet convolution, the arithmetic functions form a

commutative ring with multiplicative neutral element e where e(1) = 1 and

e(n) = 0 for n ≥ 2. This is called the Dirichlet ring. Denoting the constant

function equal to 1 by 1 we note d = 1 ∗ 1 as an example of Dirichlet

convolution. Another convolution identity is 1 ∗ Λ = log. This is easily

proved by observing that

(1 ∗ Λ)(n) =

d|n

Λ(d) =

pk|n

log(p) = log(n),

since Λ is zero off the prime powers. This can replace the Legendre identity

as the point of entry for the method of Chebyshev.

Proposition 1.12. If f and g are multiplicative, so is f ∗ g.

Proof. If gcd(m, n) = 1, then the divisors d|mn are precisely those positive

integers of the form d = bc where b|m and c|n. Hence

(f ∗ g)(mn) =

d|mn

f(d)g

mn

d

=

b|m,c|n

f(bc)g

mn

bc

=

b|m,c|n

f(b)f(c)g

m

b

g

n

c

=

b|m

f(b)g

m

b

c|n

f(c)g

n

c

= (f ∗ g)(m)(f ∗ g)(n)

by the multiplicativity of f and g.

Since 1 is multiplicative, Proposition 1.12 shows that d is also multi-

plicative. The sum-of-divisors function

σ(n) =

d|n

d

is given by the Dirichlet convolution σ = 1∗id, so σ is multiplicative, because

id is.

Part of the significance of Proposition 1.12 is that for Dirichlet con-

volutions of multiplicative functions it affords a straightforward means of

calculation; it is enough to calculate their values on prime powers. Let