Published January 20, 1997
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Numerical Studies of the Gauss Lattice Problem
 Creators
 Keller, H. B.
Abstract
The difference between the number of lattice points N(R) that lie in x^2 + y^2 ≤ R^2 and the area of that circle, d(R) = N(R)  πR^2, can be bounded by d(R) ≤ KR^θ. Gauss showed that this holds for θ = 1, but the least value for which it holds is an open problem in number theory. We have sought numerical evidence by tabulating N(R) up to R ≈ 55,000. From the convex hull bounding log d(R) versus log R we obtain the bound θ ≤ 0.575, which is significantly better than the best analytical result θ ≤ 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos.
Additional Information
[I} P.M. BIeher, Z.M. Cheng, F.J. Dyson and J.L. Lebowitz. Distribution of the error term for the number of lattice points inside a shifted circle. Comm.in Math. Phys., 154:433469, 1993. [2] J. Cizek and G. del Re. C.A. Coulson and the surface energy of metals: The distribution of eigenvalues as a difficult problem in number theory. Int. J. of Quantum Chem., 31:287293, 1987. [3] C.A. Coulson. Bull. Inst. Math. Appl., 9:2, 1973. [4] W. Fraser and C.'C. Gotlieb. A calculation of the number of lattice points in the circle and sphere. Mathematics of Computation, 16:282290, 1962. [5J C.F. Gauss. Werke, volume 2. [6] G.G. Hall. C.A. Coulson and the surfaceenergy of metals: A further comment. Int. 1. Quant., 34:301304, 1988. [7] G.H. Hardy. On Dirichlet's divisor problem. Proc. London Math. Soc., Ser. 2, 15:125, 1915. [8] D.A. Hejhal. The Selberg trace formula and the Riemann zeta function. Duke Math. J., 43:441482, 1976. [9] M.N. Huxley. Exponential sums and lattice points II. Proc. London Math. Soc., 66(2):279301, 1993. [10] H. Iwaniec and C.J. Mozzochi. On the divisor and circle problems. J. Number Theory, 29:6093, 1988. [11] L.K. Hua. The latticepoints in a circle. Quart. J. Math., Oxford Ser., 13:1829, 1942. [12] H.B. Keller and J.R. Swenson. Experiments on the lattice problem of Gauss. Mathematics of Computation, 17(83):223230, 1963. [13] G. Kolesnik. On the order of ((1/2 + it) and o(r). Pacific J. of Math., 98:107122, 1982. [14] J.E. Littlewood and A. Walfisz. Proc. of the Royal Soc., A106:478487, 1929. [15] H.L. Mitchell III. Numerical experiments on the number of lattice points in the circle. Tech. Rep. No. 17, Appl. Math. and Stat. Labs., Stanford University, 1963. [16] Nieland. Math. Ann., 98:717736,1928. [17] W. Sierpinski. Prace matematycznojizyczne, volume 17, 1906. [18] J.M. Titchmarsh. Proc. London Math. Soc. (2), 38:96115, 1935. [19] W.L. Yin. The lattice points in a circle. Scientia Sinica, 11(1):1015, 1962.Attached Files
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 Eprint ID
 16440
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 CaltechAUTHORS:20091022102132378
 CCR9120008
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 CRPC
 Series Volume or Issue Number
 971