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
