32 1. Arithmetic Functions
that log(x) + 2γ is the asymptotic law of growth of d(n) in the sense that the local
average approaches log(x)+2γ as x → +∞ and y → +∞ in a suitable way. Dirichlet
[Dir49] returned to the divisor problem in 1849 and gave his classical bound O(
for the error term. Using Dirichlet’s result L. Kronecker in his lectures [Kro01]
optimized the length of the interval to minimize the error bound for the local average
The Dirichlet divisor problem has a rich history, with a couple of obscure turns
in the early stages. The first of these is a letter from Dirichlet to Kronecker dated
July 23, 1858. Dirichlet had visited Kronecker for a few days in Ilsenburg, a resort
by the Harz mountains, where the latter was spending his summer vacation. From
the letter it is clear that they had discussed the divisor problem, and Dirichlet
writes that he has now managed to substantially improve on his result of 1849 (‘....
die Summe ... bedeutend in die Enge zu treiben.’) Dirichlet died on May 5, 1859
and his improvement never appeared. Apparently his Nachlass did not contain any
material bearing on the problem, and nothing is known of the nature of the new
method of which Dirichlet hints in the letter to Kronecker, nor of the extent of the
In 1903 G. F. Voronoi [Vor03] showed that ϑ ≤ 1/3 in the Dirichlet divisor
problem. The proof is nearly forty pages long, and is based on the interpretation
of D(x) in terms of lattice points under a hyperbola. By means of Farey frac-
tions Voronoi closely approximated the hyperbola by a polygon, and then he used
the Euler summation formula to obtain the estimate Δ(x) = O(x1/3 log(x)). The
following year he gave a different and much more analytic proof [Vor04].
As late as 1917 Voronoi’s result was judged ‘one of the deepest in the analytic
theory of numbers’ by Hardy and Ramanujan [HR17a] in their paper on the normal
number of prime factors of an integer. But in the same year I. M. Vinogradov
[Vin18a] found a much easier proof. In 1922 J. G. van der Corput [Cor22] proved
ϑ ≤ 33/100. His proof required estimates for exponential sums, and since that time
further progress has depended on better estimates for such sums and on related
techniques of counting lattice points.
The exponent in the Dirichlet divisor problem was slowly reduced over a long
period, by van der Corput (ϑ = 27/82 in 1928), T.-T. Chih [Chi50] (ϑ = 15/46
in 1950), G. A. Kolesnik [Kol69, Kol74, Kol82, Kol85] (ϑ = 12/37 in 1969,
ϑ = 346/1067 in 1973, ϑ = 35/108 in 1982 and ϑ = 139/429 in 1985), Iwaniec
and Mozzochi [IM88] (ϑ = 7/22 in 1988), and Huxley [Hux93, Hux02, Hux03]
(ϑ = 23/73 in 1993 and ϑ = 131/416 in 2000.) In the other direction, Hardy
[Har15b, Har15a] proved in 1914 that ϑ ≥ 1/4, and it is generally believed that
ϑ = 1/4 holds.
The formula of Meissel is in [Mei54]. Diamond’s version of the hyperbola
method is in his survey paper [Dia82] on elementary methods in prime number
theory. The special case y =
was noted by J. Franel [Fra99] in 1899.