Dirichlet Divisor Problem (original) (raw)
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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) |
See also
Divisor Function, Gauss's Circle Problem
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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.Vinogradov, I. M. "Anzahl der Gitterpunkte in der Kugel." Traveaux Inst. Phys.-Math. Stekloff (Leningrade) 9, 17-38, 1935. [Russian].
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Weisstein, Eric W. "Dirichlet Divisor Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletDivisorProblem.html