Sums of cubes of square-free numbers. II (Q1180711)
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scientific article; zbMATH DE number 29532
| Language | Label | Description | Also known as |
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| English | Sums of cubes of square-free numbers. II |
scientific article; zbMATH DE number 29532 |
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Sums of cubes of square-free numbers. II (English)
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27 June 1992
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[For part I, cf. ibid. 111, 1-21 (1991; Zbl 0718.11046).] Let \(R_ t(n)\) denote the number of solutions of \(x_ 1^ 3+\cdots+x_ t^ 3=n\) in square-free integers \(x_ 1,\ldots,x_ t\). We prove that \(R_ 7(n)\gg n^{4/3}\) (which is best possible) and establish an asymptotic formula for \(R_ 8(n)\). Without the restriction to square-free variables, these results are due to \textit{R. C. Vaughan} [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046), J. Lond. Math. Soc., II. Ser. 39, 205-218 (1989; Zbl 0677.10034)].
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Waring's problem
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representation of integers
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sums of cubes
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seven cubes
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eight cubes
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Vaughan's minor arc estimate
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exponential sums over square- free numbers
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mean value theorem
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square-free integers
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asymptotic formula
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