[
{
"id": "https://authors.library.caltech.edu/records/prnc0-n1f53",
"eprint_id": 16440,
"eprint_status": "archive",
"datestamp": "2023-08-19 00:59:02",
"lastmod": "2024-01-12 23:38:26",
"type": "monograph",
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"creators": {
"items": [
{
"id": "Keller-H-B",
"name": {
"family": "Keller",
"given": "H. B."
}
}
]
},
"title": "Numerical Studies of the Gauss Lattice Problem",
"ispublished": "unpub",
"full_text_status": "public",
"note": "[I} P.M. BIeher, Z.M. Cheng, F.J. Dyson and J.L. Lebowitz. Distribution\nof the error term for the number of lattice points inside a shifted circle.\nComm.in Math. Phys., 154:433-469, 1993.\n[2] J. Cizek and G. del Re. C.A. Coulson and the surface energy of metals:\nThe distribution of eigenvalues as a difficult problem in number theory.\nInt. J. of Quantum Chem., 31:287-293, 1987.\n[3] C.A. Coulson. Bull. Inst. Math. Appl., 9:2, 1973.\n[4] W. Fraser and C.'C. Gotlieb. A calculation of the number of lattice points\nin the circle and sphere. Mathematics of Computation, 16:282-290, 1962.\n[5J C.F. Gauss. Werke, volume 2.\n[6] G.G. Hall. C.A. Coulson and the surface-energy of metals: A further\ncomment. Int. 1. Quant., 34:301-304, 1988.\n[7] G.H. Hardy. On Dirichlet's divisor problem. Proc. London Math. Soc.,\nSer. 2, 15:1-25, 1915.\n[8] D.A. Hejhal. The Selberg trace formula and the Riemann zeta function.\nDuke Math. J., 43:441-482, 1976.\n[9] M.N. Huxley. Exponential sums and lattice points II. Proc. London\nMath. Soc., 66(2):279-301, 1993.\n[10] H. Iwaniec and C.J. Mozzochi. On the divisor and circle problems. J.\nNumber Theory, 29:60-93, 1988.\n[11] L.-K. Hua. The lattice-points in a circle. Quart. J. Math., Oxford Ser.,\n13:18-29, 1942.\n[12] H.B. Keller and J.R. Swenson. Experiments on the lattice problem of\nGauss. Mathematics of Computation, 17(83):223-230, 1963.\n[13] G. Kolesnik. On the order of ((1/2 + it) and o(r). Pacific J. of Math.,\n98:107-122, 1982.\n[14] J.E. Littlewood and A. Walfisz. Proc. of the Royal Soc., A106:478-487,\n1929.\n[15] H.L. Mitchell III. Numerical experiments on the number of lattice points\nin the circle. Tech. Rep. No. 17, Appl. Math. and Stat. Labs., Stanford\nUniversity, 1963.\n[16] Nieland. Math. Ann., 98:717-736,1928.\n[17] W. Sierpinski. Prace matematyczno-jizyczne, volume 17, 1906.\n[18] J.M. Titchmarsh. Proc. London Math. Soc. (2), 38:96-115, 1935.\n[19] W.-L. Yin. The lattice points in a circle. Scientia Sinica, 11(1):10-15,\n1962.\n\nPublished - CRPC-97-1.pdf

",
"abstract": "The difference between the number of lattice points N(R) that lie in x^2 + y^2 \u2264 R^2 and the area of that circle, d(R) = N(R) - \u03c0R^2, can be bounded by |d(R)| \u2264 KR^\u03b8.\n\nGauss showed that this holds for \u03b8 = 1, but the least value for which it holds is an open problem in number\ntheory. We have sought numerical evidence by tabulating N(R) up to R \u2248 55,000. From the convex hull bounding log |d(R)| versus log R we obtain the bound \u03b8 \u2264 0.575, which is significantly better than the best analytical result \u03b8 \u2264 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos.",
"date": "1997-01-20",
"date_type": "published",
"publisher": "California Institute of Technology",
"id_number": "CaltechAUTHORS:20091022-102132378",
"official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378",
"rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.",
"funders": {
"items": [
{
"agency": "NSF",
"grant_number": "CCR-9120008"
}
]
},
"collection": "CaltechAUTHORS",
"primary_object": {
"basename": "CRPC-97-1.pdf",
"url": "https://authors.library.caltech.edu/records/prnc0-n1f53/files/CRPC-97-1.pdf"
},
"resource_type": "monograph",
"pub_year": "1997",
"author_list": "Keller, H. B."
}
]