Number Theory: An Introduction to Mathematics

154 III More on Divisibility

It follows from Lemma 25 that the set of all arithmetical functions f :N→C
such thatf( 1 )=0 is an abelian group under Dirichlet multiplication.
A functionf:N→Cis said to bemultiplicativeif it is not identically zero and if

f(mn)=f(m)f(n) for allm,nwith(m,n)= 1.

It follows that f( 1 )=1, since f(n)=0forsomenand f(n)= f(n)f( 1 ).Any
multiplicative functionf:N→Cis uniquely determined by its values at the prime
powers, since if

m=pα 11 ···pαss,

wherep 1 ,...,psare distinct primes andα 1 ,...,αs∈N,then

f(m)=f(p 1 α^1 )···f(pαss).

If

m=

∏

p

pαp, n=

∏

p

pβp,

whereαp,βp≥0, then

(m,n)=

∏

p

pγp, [m,n]=

∏

p

pδp,

whereγp=min{αp,βp}andδp=max{αp,βp}. Since eitherγp=αpandδp=βp,
orγp=βpandδp=αp, it follows that, for any multiplicative function fand all
m,n∈N,

f((m,n))f([m,n])=

∏

p

f(pγp)f(pδp)=

∏

p

f(pαp)f(pβp)=f(m)f(n).

As we saw in§5 of Chapter II, it follows from Proposition II.4 that Euler’s
φ-function is multiplicative. Also, the functionsi:N→Candj:N→C,definedby

i(n)= 1 ,j(n)=n for everyn∈N,

are obviously multiplicative. Further examples of multiplicative functions can be con-
structed with the aid of the next two propositions.

Proposition 26If f,g:N→Care multiplicative functions, then their Dirichlet
product h=f∗g is also multiplicative.

Proof We h av e

h(n)=

∑

d|n

f(d)g(n/d).

Supposen=n′n′′,wheren′andn′′are relatively prime. Then, by Proposition II.4,