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The asymptotic formula in Waring’s problem. (English) Zbl 0815.11050

As usual, let \(r_{k,s} (n)\) denote the number of representations of \(n\) as a sum \(x_ 1^ k + x_ 2^ k + \cdots + x_ s^ k\) of positive integers. Let \(\widetilde G(k)\) denote the least \(s_ 0\) such that for all \(s \geq s_ 0\) the asymptotic formula of Hardy and Littlewood \[ r_{s,k} (n) = {\mathfrak S}_{s,k} (n) f(k) n^{(s/k) - 1} \bigl( 1+ o(1) \bigr) \] holds. Here \({\mathfrak S}_{s,k} (n)\) denotes the usual “singular series” and \(f(k)\) a certain product of \(\Gamma\)-functions. Following observations made by Hardy and Littlewood themselves, the known bounds on \(\widetilde G(k)\) are not the same as those on the more well- known number \(G(k)\), the least \(s_ 0\) such that a representation of the desired form exists for all sufficiently large \(n\).
This paper improves the bounds for \(\widetilde G(k)\) when \(k\) is of moderate size, \(6 \leq k \leq 9\). R. Vaughan [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046), Mathematika 33, 6-22 (1986; Zbl 0601.10037)] showed \(\widetilde G(k) \leq 2^ k\) when \(k \geq 3\). By combining ideas due to L.-K. Hua and I. M. Vinogradov, D. R. Heath- Brown [J. Lond. Math. Soc., II. Ser. 38, 216-230 (1988; Zbl 0657.10051)] showed by methods independent of those of Vaughan that \(\widetilde G (k) \leq {7 \over 8} 2^ k + 1\) when \(k \geq 6\). By combining the methods of Heath-Brown and of Vaughan the author’s analysis reduces Heath-Brown’s bound by 1.
For larger \(k\) sharper bounds on \(\widetilde G(k)\) follow from improvements on the methods of I. M. Vinogradov made by T. D. Wooley [Mathematika 39, 379-399 (1992; Zbl 0769.11036)] who (approximately halving the previous bounds) established \(\widetilde G(k) < (2 + o(1))k^ 2 \log k\) in general, and \(\widetilde G(10) \leq 750\) and other explicit results in particular. A further halving of Wooley’s asymptotic bound and an improvement in the bound on \(\widetilde G(k)\) for \(k \geq 9\) is expected in forthcoming work by K. Ford.

MSC:

11P55 Applications of the Hardy-Littlewood method
11P05 Waring’s problem and variants
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References:

[1] Wooley, Mathematika 39 pp 379– (1992)
[2] Vaughan, Mathematika 33 pp 6– (1986)
[3] Vaughan, J. reine angew. Math 365 pp 122– (1986)
[4] Hall, Compositio Math 60 pp 163– (1986)
[5] DOI: 10.1093/qmath/os-9.1.199 · Zbl 0020.10504
[6] DOI: 10.1112/jlms/s2-38.2.216 · Zbl 0619.10046
[7] DOI: 10.1007/BF01482074 · JFM 48.0146.01
[8] Vaughan, The Hardy-Littlewood Method (1981)
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