## On Waring’s problem for smaller exponents. II.(English)Zbl 0601.10037

[For part I, cf. Proc. Lond. Math. Soc., III. Ser. 52, 445-463 (1986; Zbl 0601.10035).]
The asymptotic formula for the number of representations of a (large) integer $$N$$ as sums of $$s$$ $$k$$-th powers ($$k\geq 2)$$ was established by Hua Lookeng [Q. J. Math., Oxf. Ser. 9, 199-202 (1938; Zbl 0020.10504)] with $$s\geq 2^ k+1$$. Vinogradov’s results are much stronger than this for large $$k$$. However, for $$3\leq k\leq 10$$, Hua’s result has been the best known result.
In this paper, the author shows that the asymptotic formula holds with $$s\geq 2^ k$$. With $$P=N^{1/k}$$, $$f(\alpha)=\sum_{1\leq x\leq P}e(\alpha x^ k)$$, Hua’s inequality asserts that $\int^{1}_{0}| f(\alpha)|^{2^ k} d\alpha \ll P^{2^ k- k+\epsilon}$ (and more precisely, $$P^{\epsilon}$$ can be replaced by $$(\log P)^ C$$ for some positive constant $$C$$). Thus, in using the Hardy-Littlewood method for establishing the asymptotic formula, Hua uses Weyl’s inequality for an additional $$k$$-th power in order to save more than $$P^{-k}$$ over the minor arcs. With $$m$$ denoting the minor arcs, the author shows that $\int_{m}| f(\alpha)|^{2^ k} d\alpha \ll P^{2^ k-k}\quad (\log P)^{-2+\epsilon},$ and consequently, does not require the additional $$k$$-th power for the minor arcs (the major arcs presenting no problem).
Reviewer: K.Thanigasalam

### MSC:

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

### Citations:

Zbl 0601.10035; Zbl 0020.10504
Full Text:

### References:

 [1] Halberstam, Sieve Methods (1974) [2] DOI: 10.1007/BF01111351 · Zbl 0151.03802 [3] van der Corput, Ned. Akad. Wet. Proc. 42 pp 547– (1939) [4] DOI: 10.2307/1968889 · Zbl 0024.01402 [5] DOI: 10.1007/BF01351892 · Zbl 0104.04201 [6] DOI: 10.1007/BF01482074 · JFM 48.0146.01 [7] Vaughan, The Hardy-Littlewood Method (1981) [8] DOI: 10.1093/qmath/os-9.1.199 · Zbl 0020.10504 [9] Hua, Additive Theory of Prime Numbers, A.M.S. translations of mathematical monographs (1965) [10] DOI: 10.1112/plms/s3-43.1.73 · Zbl 0463.10035 [11] DOI: 10.1112/plms/s3-38.1.115 · Zbl 0394.10027 [12] Vaughan, Coll. Math. Soc. János Bolyai, Budapest pp 1585– (1981)
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