Lemma
(Dirichlet hyperbola method)
.
Assuming that:
f
,
g
:
ℕ
→
ℂ
x
≥
y
≥
1
Then
∑
n
≤
x
(
f
∗
g
)
(
n
)
=
∑
d
≤
y
f
(
d
)
∑
m
≤
x
d
g
(
m
)
+
∑
m
≤
x
y
g
(
m
)
∑
y
<
d
≤
x
m
f
(
d
)
.