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On Waring's problem for cubes and biquadrates. II

  • Jörg Brüdern (a1)


In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.



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[1]Brüdern, J.. On Waring's problem for cubes and biquadrates. J. London Math. Soc. (2) 37 (1988), 2542.
[2]Brüdern, J.. A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 2733.
[3]Davenport, H.. On sums of positive integral k-th powers. Amer. J. Math. 64 (1942), 189198.
[4]Hooley, C.. On Waring's problem. Acta Math. 157 (1986), 4997.
[5]Thanigasalam, K.. On sums of mixed powers. Bull. Calcutta Math. Soc. 77 (1985), 1719.
[6]Vaughan, R. C.. The Hardy–Littlewood Method (Cambridge University Press, 1981).
[7]Vaughan, R. C.. On Waring's problem for cubes. J. reine angew. Math. 365 (1986), 122170.
[8]Vaughan, R. C.. On Waring's problem: one square and five cubes. Quart. J. Math. Oxford Ser. (2) 37 (1986), 117127.
[9]Vaughan, R. C.. On Waring's problem for smaller exponents. Proc. London Math. Soc. (3) 52 (1986), 445463.

On Waring's problem for cubes and biquadrates. II

  • Jörg Brüdern (a1)


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