“Gauss's Circle Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gauss's Circle Problem

释义

Gauss's Circle Problem

Count the number of Lattice Points inside the boundary of a Circle ofRadius with center at the origin. The exact solution is given by the Sum

(1)

The first few values for

, 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (Sloane's A000328).

Gauß showed that

(2)

where

(3)

Writing

, the best bounds on

are

(Huxley 1990). Theproblem has also been extended to Conics and higher dimensions. The limit 1/2 was obtained byHardy and Landau (1915), and the limit 46/73 improves previous values of

(Cheng 1963) and

(Vinogradov), and

See also Circle Lattice Points

References

Cheng, 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.

Guy, R. K. ``Gauß's Lattice Point Problem.'' §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240-2417, 1994.

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.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983.

Sloane, N. J. A. SequenceA000328/M3829in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weisstein, E. W. ``Circle Lattice Points.'' Mathematica notebook CircleLatticePoints.m.

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