The Dirichlet divisor problem asks for bounds on the error in the asymptotic formula for the summatory divisor function. Let
be the number of divisors of the positive
integer
(including
itself). For a prime, . In general,
where
is the Euler-Mascheroni constant. Dirichlet
originally gave (Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68),
and Hardy and Landau showed in 1916 that (Hardy 1999, p. 81).
Bourgain and Watt (2017) claimed the bound for every , but N. Watt withdrew the paper on July 16,
2023 because the step bounding expression (3.9) by expression (3.11) in the proofs
of Propositions 2 and 3 was not justified.
Li and Yang (2023, p. 2) report that the error is for every , with . Their result applies to both this
problem and Gauss's circle problem.
The following table summarizes historical improvements through 2003 (updated from Hardy 1999, p. 81).
approx.
citation
1/2
0.50000
Dirichlet
1/3
0.33333
Voronoi
(1903), Sierpiński (1906), van der Corput (1923)
Bohr, H. and Cramér, H. "Ellipsoidprobleme." In "Die neuere Entwicklung der analytischen Zahlentheorie." Ch. IIC88
in Enzykl. d. Math. Wiss., Vol. 2, Part 3, Issue 2II C 8, 823-824,
1922.Bourgain, J. and Watt, N. "Mean Square of Zeta Function, Circle
Problem and Divisor Problem Revisited." 13 Sep 2017. Withdrawn 16 Jul 2023.
https://arxiv.org/abs/1709.04340.Chen,
J.-R. "The Lattice-Points in a Circle." Sci. Sinica12, 633-649,
1963.Graham, S. W. and Kolesnik, G. Van
Der Corput's Method of Exponential Sums. Cambridge, England: Cambridge University
Press, 1991.Hardy, G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, 1999.Hardy, G. H. and Wright, E. M. An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.Huxley, M. N. "Exponential Sums and Lattice Points."
Proc. London Math. Soc.60, 471-502, 1990.Huxley, M. N.
"Corrigenda: 'Exponential Sums and Lattice Points.' " Proc. London Math.
Soc.66, 70, 1993.Huxley, M. N. "Exponential Sums
and Lattice Points. II." Proc. London Math. Soc.66, 279-301,
1993.Huxley, M. N. "Exponential Sums and Lattice Points III."
Proc. London Math. Soc.87, 591-609, 2003. https://doi.org/10.1112/S0024611503014485.Iwaniec,
H. and Mozzochi, C. J. "On the Divisor and Circle Problems." J.
Numb. Th.29, 60-93, 1988.Kolesnik, G. "On the Order
of
and ."
Pacific J. Math.98, 107-122, 1982.Kolesnik, G. A.
"An Improvement of the Remainder Term in the Divisor Problem." Mat.
Zametki6, 545-554, 1969. English translation in Math. Notes6,
784-791, 1969.Li, X. and Yang, X. "An Improvement on Gauss's Circle
Problem and Dirichlet's Divisor Problem." 14 Sep 2023. https://arxiv.org/abs/2308.14859.Littlewood,
J. E. and Walfisz, A. "The Lattice Points of a Circle. (With a Note by
Prof. E. Landau.)." Proc. Roy. Soc. London (A)106, 478-488,
1925.van der Corput, J. G. "Zum Teilerproblem." Math.
Ann.98, 697-716, 1928.Vinogradov, I. M. "Anzahl
der Gitterpunkte in der Kugel." Traveaux Inst. Phys.-Math. Stekloff (Leningrade)9,
17-38, 1935. [Russian].
be the number of divisors of the positive
integer 
(including 
itself). For a prime 
, 
. In general,
is the Euler-Mascheroni constant. Dirichlet
originally gave 
(Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68),
and Hardy and Landau showed in 1916 that 
(Hardy 1999, p. 81).
for every 
, but N. Watt withdrew the paper on July 16,
2023 because the step bounding expression (3.9) by expression (3.11) in the proofs
of Propositions 2 and 3 was not justified.
for every 
, with 
. Their result applies to both this
problem and Gauss's circle problem.
and 
."
Pacific J. Math. 98, 107-122, 1982.Kolesnik, G. A.
"An Improvement of the Remainder Term in the Divisor Problem." Mat.
Zametki 6, 545-554, 1969. English translation in Math. Notes 6,
784-791, 1969.Li, X. and Yang, X. "An Improvement on Gauss's Circle
Problem and Dirichlet's Divisor Problem." 14 Sep 2023.