1.6. The hyperbola method 25
asymptotic estimate for the local average may be improved in two different
ways: By shortening the interval or reducing the error term. Modifying the
estimate as it stands by shortening the interval as much as possible, we lose
the error term and obtain the asymptotic estimate
1
h
x−hn≤x
d(n) log(x) + 2γ, h =
o(x1/2/
log(x)).
Using a better estimate in the divisor problem, the estimate for the local
average may be improved both by reducing the error term and shortening
the interval.
One may also prove Proposition 1.16 from Dirichlet’s formula
D(x) =
dk≤x
1 =
n≤x
x
n
.
That D(x) = x log(x) + O(x) is immediate from this formula. The better
estimate for the error term may be obtained by observing that the sum
equals the number of integer lattice points in the region of the uv-plane
given by the inequalities u 1, v 1 and uv x. One can then recover
the formula of Meissel by observing that the union of the two subregions
obtained by imposing the inequalities u

x and v

x equals the
region, while their intersection equals the square given by 1 u
√original
x and
1 v

x. The interpretation of D(x) in terms of the number of lattice
points under a hyperbola is very important for more advanced work on the
divisor problem. See Figure 2 on page 26 for an illustration.
The Dirichlet interchange and the approach to the Meissel formula based
on counting lattice points under a hyperbola are closely connected. The
technique is usually called the Dirichlet hyperbola method. It has other
applications and so we exhibit a more general formulation due to H. G.
Diamond.
Proposition 1.17 (Dirichlet hyperbola method). If f is an arithmetic func-
tion with summatory function F and g an arithmetic function with summa-
tory function G then
n≤x
(f g)(n) =
k≤y
f(k)G
x
k
+
m≤x/y
g(m)F
x
m
F (y)G
x
y
for 1 y x.
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