Dirichlet Divisor Problem
Let the divisor function 
be the number of divisors of 
(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). The following table summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).
 |
approx. |
citation |
| 1/2 |
0.50000 |
Dirichlet |
| 1/3 |
0.33333 |
Voronoi (1903), Sierpiński (1906), van der Corput (1923) |
| 37/112 |
0.33036 |
Littlewood and Walfisz (1925) |
| 33/100 |
0.33000 |
van der Corput (1922) |
| 27/82 |
0.32927 |
van der Corput (1928) |
| 15/46 |
0.32609 |
|
| 12/37 |
0.32432 |
Chen (1963), Kolesnik (1969) |
| 35/108 |
0.32407 |
Kolesnik (1982) |
| 139/429 |
0.32401 |
Kolesnik |
| 17/53 |
0.32075 |
Vinogradov (1935) |
| 7/22 |
0.31818 |
Iwaniec and Mozzochi (1988) |
| 23/73 |
0.31507 |
Huxley (1993) |
| 131/416 |
0.31490 |
Huxley (2003) |
REFERENCES:
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 2 II C 8, 823-824, 1922.
Chen, J.-R. "The Lattice-Points in a Circle." Sci. Sinica 12, 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, 5910-609, 2003.
Iwaniec, H. and Mozzochi, C. J. "On the Divisor and Circle Problem." J. Numb. Th. 29, 60-93, 1988.
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.
Kolesnik, G. "On the Order of 
and 
." Pacific J. Math. 98, 107-122, 1982.
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.
