Number Theory: An Introduction to Mathematics

152 III More on Divisibility

1993 Fermat’s assertion had been established in this way for allnless than four million.
However, these methods did not lead to a complete proof of ‘Fermat’s last theorem’.
As will be seen in Chapter XIII, a complete solution was first found by Wiles (1995),
using quite different methods.

3 Multiplicative Functions

We define a functionf :N→Cto be anarithmetical function. The set of all arith-
metical functions can be given the structure of a commutative ring in the following
way.
For any two functions f,g :N→C, we define theirconvolutionorDirichlet
product f∗g:N→Cby

f∗g(n)=

∑

d|n

f(d)g(n/d).

Dirichlet multiplication satisfies the usual commutative and associative laws:

Lemma 24For any three functions f,g,h:N→C,

f∗g=g∗f, f∗(g∗h)=(f∗g)∗h.

Proof Sincen/druns through the positive divisors ofnat the same time asd,

f∗g(n)=

∑

d|n

f(d)g(n/d)

=

∑

d|n

f(n/d)g(d)=g∗f(n).

To prove the associative law, putG=g∗h.Then

f∗G(n)=

∑

de=n

f(d)G(e)=

∑

de=n

f(d)

∑

d′d′′=e

g(d′)h(d′′)

=

∑

dd′d′′=n

f(d)g(d′)h(d′′).

Similarly, if we putF=f∗g, we obtain

F∗h(n)=

∑

de=n

F(e)h(d)=

∑

de=n

∑

d′d′′=e

f(d′)g(d′′)h(d)

=

∑

dd′d′′=n

f(d′)g(d′′)h(d).

HenceF∗h(n)=f∗G(n).