Gauss's Circle Problem

Gauss's circle problem asks for the number of lattice points on or inside a circle of radius centered at the origin. The exact solution is given by the sum

			(1)
			(2)
			(3)

where is the floor function (Hilbert and Cohn-Vossen 1999, p. 39). The first few values for , 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (OEIS A000328).

The series for is intimately connected with the sum of squares function (i.e., the number of representations of by two squares), since

(4)

(Hardy 1999, p. 67). The Leibniz series gives

(5)

(Hilbert and Cohn-Vossen 1999, p. 39). The limit of as is .

Gauss showed that

(6)

where

(7)

for (Hardy 1999, p. 67).

The first few values of are 5, 13/4, 29/9, 49/16, 81/25, 113/36, 149/49, 197/64, 253/81, 317/100, 377/121, 49/16, ... (OEIS A093836 and A093837). As can be seen in the plot above, the values of such that are , 2, 3, 5, 10, 15, 20, 35, 51, 52, 85, 100, ... (OEIS A093832).

Let be the infimum of the exponents such that for every . The lower bound of Hardy (1915) and Landau (1915) and Huxley's (2003) upper bound give

(8)

The exponent in Huxley's bound is accompanied by a power of , which is absorbed by . Li and Yang (2023, p. 2) report an improved upper bound .

Cen et al. (2026, p. 1) report the stronger bound

(9)

for every and . Their proof has not been independently verified here. The authors report ChatGPT and Codex assistance with proof strategies and checking parts of the mathematical arguments.

The following table summarizes historical improvements through 2003 (updated from Hardy 1999, p. 81).

	approx.	citation
1	1.00000	Dirichlet
2/3	0.66667	Voronoi (1903), Sierpiński (1906), van der Corput (1923)
37/56	0.66071	Littlewood and Walfisz (1925)
33/50	0.66000	van der Corput (1922)
27/41	0.65854	van der Corput (1928)
15/23	0.65217
24/37	0.64865	Chen (1963), Kolesnik (1969)
35/54	0.64815	Kolesnik (1982)
278/429	0.64802	Kolesnik
34/53	0.64151	Vinogradov (1935)
7/11	0.63636	Iwaniec and Mozzochi (1988)
46/73	0.63014	Huxley (1993)
131/208	0.62981	Huxley (2003)

The problem has also been extended to conics, ellipsoids (Hardy 1915), and higher dimensions.

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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.Cen, X.; Gao, X.; Zhang, C.; and Zhang, J. "A New Bound for the Gauss Circle Problem." 15 Oct 2026. https://arxiv.org/abs/2610.21730.Chen, J.-R. "The Lattice-Points in a Circle." Sci. Sinica 12, 633-649, 1963.Cilleruello, J. "The Distribution of Lattice Points on Circles." J. Number Th. 43, 198-202, 1993.Graham, S. W. and Kolesnik, G. Van Der Corput's Method of Exponential Sums. Cambridge, England: Cambridge University Press, 1991.Guy, R. K. "Gauß's Lattice Point Problem." §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240-241, 1994.Hardy, G. H. "On the Expression of a Number as the Sum of Two Squares." Quart. J. Math. 46, 263-283, 1915.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, pp. 268-269, 1979.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, pp. 33-35, 1999.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.Iwaniec, H. and Mozzochi, C. J. "On the Divisor and Circle Problem." J. Numb. Th. 29, 60-93, 1988.Keller, H. B. and Swenson, J. R. "Experiments on the Lattice Problem of Gauss." Math. Comput. 17, 223-230, 1963.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. Zametki 6, 545-554, 1969. English translation in Math. Notes 6, 784-791, 1969.Landau, E. "Über die Anzahl der Gitterpunkte in gewissen Bereichen. I." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 687-770, 1912.Landau, E. "Neue Untersuchungen über die Pfeiffer'sche Methode zur Abschätzung von Gitterpunktanzahlen." Sitzungsber. d. math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2. Abteilung, Wien, No. 124, 469-505, 1915.Landau, E. "Über die Gitterpunkte in einem mehrdimensionalen Ellipsoid." In Zur analytischen Zahlentheorie der definiten quadratischen Formen. Sitzungsber. d. Berliner math. Gesellschaft, 458-476, 1915.Landau, E. "Über die Anzahl der Gitterpunkte in gewissen Bereichen. II." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 209-243, 1915.Landau, E. "Über die Gitterpunkte in einem Kreise. I." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 148-160, 1915.Landau, E. "Über die Gitterpunkte in einem Kreise. II." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 161-171, 1915.Landau, E. "Über die Anzahl der Gitterpunkte in gewissen Bereichen. III." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 96-101, 1917.Landau, E. "Über die Gitterpunkte in einem Kreise. III." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 109-134,1920.Landau, E. "Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 137-150, 1924.Landau, E. "Über die Gitterpunkte in einem Kreise. IV." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse, 58-65, 1924.Landau, E. "Über die Gitterpunkte in einem Kreise. V." Nachr. v. d. Gesellschaft d. Wiss. zu Göttingen, math.-phys. Klasse 135-136, 1924.Landau, E. "Über Gitterpunkte in mehrdimensionalen Ellipsoiden. I." Math. Z. 21, 126-132, 1924.Landau, E. "Über Gitterpunkte in mehrdimensionalen Ellipsoiden. II." Math. Z. 24, 299-310, 1925.Landau, E. Vorlesungen über Zahlentheorie, Vol. 2. New York: Chelsea, pp. 183-308, 1970.Landau, E. and van der Corput, J. G. "Über Gitterpunkte in ebenen Bereichen." Nachr. v. d. Gesellschaft d. Wissenschaften zu Göttingen, math.-phys. Klasse, 135-171, 1920.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 24, 1983.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.Sloane, N. J. A. Sequences A000328/M3829, A093832, A093836, and A093837 in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh, E. C. "On van der Corput's Method and the Zeta-Function of Riemann. I." Quart. J. Math. (Oxford) 2, 161-173, 1931.Titchmarsh, E. C. "On van der Corput's Method and the Zeta-Function of Riemann. II." Quart. J. Math. (Oxford) 2, 313-320, 1931.Titchmarsh, E. C. "The Lattice Points in a Circle." Proc. London Math. Soc. 28, 96-115, 1934.Titchmarsh, E. C. "Corrigendum. The Lattice-Points in a Circle." Proc. London Math. Soc. 38, 555, 1935.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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Gauss's Circle Problem

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Weisstein, Eric W. "Gauss's Circle Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GausssCircleProblem.html

Gauss's Circle Problem

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