Dirichlet Divisor Problem

Let the divisor function d(n) be the number of divisors of n (including n itself). For a prime p, d(p)=2. In general,


where gamma is the Euler-Mascheroni constant. Dirichlet originally gave theta approx 1/2 (Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68), and Hardy and Landau showed in 1916 that theta>=1/4 (Hardy 1999, p. 81). The following table summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).

1/30.33333Voronoi (1903), Sierpiński (1906), van der Corput (1923)
37/1120.33036Littlewood and Walfisz (1925)
33/1000.33000van der Corput (1922)
27/820.32927van der Corput (1928)
12/370.32432Chen (1963), Kolesnik (1969)
35/1080.32407Kolesnik (1982)
17/530.32075Vinogradov (1935)
7/220.31818Iwaniec and Mozzochi (1988)
23/730.31507Huxley (1993)
131/4160.31490Huxley (2003)

See also

Divisor Function, Gauss's Circle Problem

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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 zeta(1/2+it) and Delta(R)." 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.Vinogradov, I. M. "Anzahl der Gitterpunkte in der Kugel." Traveaux Inst. Phys.-Math. Stekloff (Leningrade) 9, 17-38, 1935. [Russian].

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Dirichlet Divisor Problem

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Weisstein, Eric W. "Dirichlet Divisor Problem." From MathWorld--A Wolfram Web Resource.

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