Keller, H. B. (1997) Numerical Studies of the Gauss Lattice Problem. CRPC, 97-1. California Institute of Technology , Pasadena, CA. http://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378
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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.
|Item Type:||Report or Paper (Technical Report)|
|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., 154:433-469, 1993.  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., 31:287-293, 1987.  C.A. Coulson. Bull. Inst. Math. Appl., 9:2, 1973.  W. Fraser and C.'C. Gotlieb. A calculation of the number of lattice points in the circle and sphere. Mathematics of Computation, 16:282-290, 1962. [5J C.F. Gauss. Werke, volume 2.  G.G. Hall. C.A. Coulson and the surface-energy of metals: A further comment. Int. 1. Quant., 34:301-304, 1988.  G.H. Hardy. On Dirichlet's divisor problem. Proc. London Math. Soc., Ser. 2, 15:1-25, 1915.  D.A. Hejhal. The Selberg trace formula and the Riemann zeta function. Duke Math. J., 43:441-482, 1976.  M.N. Huxley. Exponential sums and lattice points II. Proc. London Math. Soc., 66(2):279-301, 1993.  H. Iwaniec and C.J. Mozzochi. On the divisor and circle problems. J. Number Theory, 29:60-93, 1988.  L.-K. Hua. The lattice-points in a circle. Quart. J. Math., Oxford Ser., 13:18-29, 1942.  H.B. Keller and J.R. Swenson. Experiments on the lattice problem of Gauss. Mathematics of Computation, 17(83):223-230, 1963.  G. Kolesnik. On the order of ((1/2 + it) and o(r). Pacific J. of Math., 98:107-122, 1982.  J.E. Littlewood and A. Walfisz. Proc. of the Royal Soc., A106:478-487, 1929.  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.  Nieland. Math. Ann., 98:717-736,1928.  W. Sierpinski. Prace matematyczno-jizyczne, volume 17, 1906.  J.M. Titchmarsh. Proc. London Math. Soc. (2), 38:96-115, 1935.  W.-L. Yin. The lattice points in a circle. Scientia Sinica, 11(1):10-15, 1962.|
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|Deposited By:||Kristin Buxton|
|Deposited On:||27 Oct 2009 18:45|
|Last Modified:||26 Dec 2012 11:30|
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