Concrete mathematics : a foundation for computer science

7.7 DIRICHLET GENERATING FUNCTIONS 357
Dirichlet generating functions are particularly valuable when the sequence
(gl,g2,...) is a multiplicative function, namely when
gmn = gm gn for m I n.
In such cases the v,alues of gn for all n are determined by the values of g,, when
n is a power of a prime, and we can factor the dgf into a product over primes:
G(z) = I-I
p prime ( ,+!E+w+!!!?L+...
PLZ P3= >
If, for instance, we set gn = 1 for all n, we obtain a product representation
of Riemann’s zeta function:
L(z) = p gm,.( &) ’
The Mobius function has v(p) = -1 and p(pk) = 0 for k > 1, hence its dgf is
G(z) = n ( 1 -p-“); (7.91)
p prime
this agrees, of course, with (7.88) and (7.90). Euler’s cp function has cp(pk) =
Pk-P k-’ , hence its dgf has the factored form
TNe conclude that g(z) = I(z - l)/