24 1. Arithmetic Functions
Proof. We have
D(x) = 2
n≤

x
x
n
[

x]2
= 2
n≤

x
x
n
+ O(1) (

x +
O(1))2
= 2x(log(

x) + γ + O(1/

x)) x +
O(x1/2)
= x log(x) + (2γ 1)x +
O(x1/2)
by Proposition 1.7.
This estimate is due to J. P. G. Lejeune Dirichlet. It implies that the
arithmetic average of d(n) over the range 1 n x is asymptotic to log(x)
as x +∞. One says that d(n) has average order log(x). From Proposition
1.15 it is easy to see that d(n) is sometimes larger than any fixed power of
log(n). But Proposition 1.16 implies that d(n) is only rarely so large. The
notation Δ(x) = D(x) x log(x) (2γ 1)x is traditional for the error
term in the estimate in Proposition 1.16. The problem of bounding Δ(x) is
known as the Dirichlet divisor problem. More precisely, the divisor problem
is to find the least ϑ for which an estimate Δ(x) =
O(xϑ+ε)
holds for all
ε 0. The result just proved shows that ϑ 1/2. For x large and h rather
smaller than x, say h x/2, one obtains
1
h
x−hn≤x
d(n) =
D(x) D(x h)
h
=
x log(x) + (2γ 1)x + Δ(x)
h

(x h) log(x h) + (2γ 1)(x h) + Δ(x h)
h
= log(x) + + O
h
x
+ O
x1/2
h
by the estimate Δ(x) =
O(x1/2).
The error is a sum of two terms, one of
which dominates when h is large and the other when h is small. In such
situations one would usually try to choose the parameter optimally to obtain
a small error term overall. Minimizing h/x +
x1/2/h
over h for x fixed, one
sees that h =
x3/4
is an optimal choice. Thus
1
h
x−hn≤x
d(n) = log(x) + +
O(x−1/4),
h =
x3/4.
The asymptotic law of growth log(x) + for the local average of d(n) was
Dirichlet’s main application in his 1849 paper on the divisor problem. The
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