H. B. Keller, J. R. Swenson, Experiments on the lattice problem of Gauss, Mathematics of Computation 17 (1963), 223-230.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R is the magnification factor of the unit curve) 4 H. B. KELLER 2. NUMERICAL APPROACH Since all the rigorous bounds are relatively far from the ideal 1 2 # conjectured by Hardy, it is perhaps of some interest to obtain numerical evidence of the behavior of d(R) This has been done [4,15,12,1] but with no suggested improvements. These previous attempts sampled d(R) at various sets of uniformly spaced values R j . But as d(R) is not a well behaved function of R, such samplings do not yield any useful information. Indeed N(R) is a piecewise constant function of R, continuous from ....

....enlarged set. 3. COUNTING PROCEDURES Two of the previous attempts to compute d(R) contain serious errors. In [15] the square root is fit by a table in order to speed up the calculations. But the table contains an error and thus the results for R 3000 are incorrect. This error was reported in [12]. The most recent work, by Bleher, Cheng, Dyson and Lebowitz [1] containing Tables 1a of min d(r) # r r R and 1b of max d(r) # r r R cannot be correct. Simply note that d(1) 1 = 1 # = 2.1415928 . However, all the entries in Table 1a for R 2 # 3025 have entries ....

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H.B. Keller and J.R. Swenson. Experiments on the lattice problem of Gauss. Mathematics of Computation, 17(83):223-230, 1963.

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H. B. Keller, J. R. Swenson, Experiments on the lattice problem of Gauss, Mathematics of Computation 17 (1963), 223-230.

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