## 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

### Citations:

Zbl 0574.10046; Zbl 0601.10037; Zbl 0657.10051; Zbl 0769.11036
Full Text:

### 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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