Concrete mathematics : a foundation for computer science

and we can group these fractions by their denominators:
4.9 PHI AND MU 135
What can we make of this? Well, every divisor d of 12 occurs as a denominator,
together with all cp(d) of its numerators. The only denominators that
occur are divisors of 12. Thus
dl) + (~(2) + (~(3) + (~(4) + (~(6) + (~(12) = 12.
A similar thing will obviously happen if we begin with the unreduced fractions
0 1
rn, ;;;I . . . . y for any m, hence
xv(d) = m.
d\m
(4.54)
We said near the beginning of this chapter that problems in number
theory often require sums over the divisors of a number. Well, (4.54) is one
such sum, so our claim is vindicated. (We will see other examples.)
Now here’s a curious fact: If f is any function such that the sum
g(m) = x+(d)
d\m
is multiplicative, then f itself is multiplicative. (This result, together with
(4.54) and the fact that g(m) = m is obviously multiplicative, gives another
reason why cp(m) is multiplicative.) We can prove this curious fact by induction
on m: The basis is easy because f (1) = g (1) = 1. Let m > 1, and
assume that f (ml m2) = f (ml ) f (mz) whenever ml I mz and ml mz < m. If
m=mlmz andml Imz,wehave
g(mlm) = t f(d) = t x f(dldz),
d\ml m2 dl\ml dz\mz
and dl I d2 since all divisors of ml are relatively prime to all divisors of
ml. By the induction hypothesis, f (dl d2) = f (dl ) f (dr ) except possibly when
dl = ml and d2 = m2; hence we obtain
( t f(dl) t f(b)) - f(m)f(w) + f(mmz)
dl \ml dz\m
= s(ml)s(mz) -f(ml)f(m2) +f(mm2).
But this equals g(mlmr) = g(ml)g(mz), so f(mlm2) = f(ml)f(mr).