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.