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)
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.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
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"Corrigenda: 'Exponential Sums and Lattice Points.' " Proc. London Math.
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and Lattice Points. II." Proc. London Math. Soc.66, 279-301,
1993.Huxley, M. N. "Exponential Sums and Lattice Points III."
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Th.29, 60-93, 1988.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.Kolesnik,
G. "On the Order of and ." Pacific J. Math.98, 107-122,
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of a Circle. (With a Note by Prof. E. Landau.)." Proc. Roy. Soc. London
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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
(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). The following table
summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).
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].