26 1. Arithmetic Functions
x
x
uv x
u
v
Figure 2. Lattice points in the divisor problem
Proof. We have
n≤x
(f ∗ g)(n) =
n≤x km=n
f(k)g(m)
=
k≤y
f(k)
km≤x
g(m) +
ky km≤x
f(k)g(m)
=
k≤y
f(k)G
x
k
+
m≤x/y
g(m)
yk≤x/m
f(k)
=
k≤y
f(k)G
x
k
+
m≤x/y
g(m)⎝
⎛
k≤x/m
f(k) −
k≤y
f(k)⎠
⎞
=
k≤y
f(k)G
x
k
+
m≤x/y
g(m)F
x
m
− F (y)G
x
y
for any y with 1 ≤ y ≤ x.
Questions about divisors more delicate than their gross count have also
been studied. The monograph Divisors by R. R. Hall and G. Tenenbaum is
a good source for information of this kind.
