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.


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