1.6. The hyperbola method 23

Another approach to study the growth of the rapidly fluctuating arith-

metic function d(n) is to consider a local average such as

1

h

x−hn≤x

d(n).

The fluctuations of d(n) are smoothed out by the process of averaging. Then

1

h

x−hn≤x

d(n) =

D(x) − D(x − h)

h

where D(x) is the summatory function of the divisor function. The use of

local averaging to study the growth of arithmetic functions can be traced

back to article 301 in the Disquisitiones Arithmeticae. Gauss was interested

in the growth of the class number and the number of genera for binary

quadratic forms, as (irregularly fluctuating) arithmetic functions of the dis-

criminant. He quoted results on the rate of growth of their local averages,

but judged the proofs to be too diﬃcult to include in the Disquisitiones.

To calculate a local average by differencing an estimate for the associated

summatory function is not always eﬃcient. When h is small, one is apt to

run into the same kind of problem as one does in numerics when subtracting

floating-point numbers that are nearly equal.

The bijection d → n/d on the set of divisors d of an integer n is called

the Dirichlet interchange. Since d

√

n is equivalent to n/d

√

n it is

clear that

d(n) = 2

√

nd|n

1

unless n is a square. In the latter case the divisor

√

n is missing and it is

necessary to add 1 on the right-hand side. Applying this formula to the

definition of D(x) gives

D(x) = [

√

x] +

n≤x

2

√

nd|n

1 = [

√

x] + 2

d≤

√

x d2kd≤x

1

= [

√

x] + 2

d≤

√

x

x

d

− d = 2

m≤

√

x

x

m

− [

√

x]2.

The latter formula is due to D. F. E. Meissel.

Proposition 1.16. The estimate

D(x) = x log(x) + (2γ − 1)x +

O(x1/2)

holds.