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ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS

  • JÖRG BRÜDERN (a1)

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1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation (1)

$ \begin{equation*} r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty). \end{equation*} $
Here ${\mathfrak s}$ (n) is the singular series associated with sums of a square and three fifth powers, see (13) below for a precise definition. The main purpose of this note is to confirm (1) in mean square.

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1.Brüdern, J., Sums of squares and higher powers. I. J. London Math. Soc. (2) 35 (2) (1987), 233243.
2.Brüdern, J., A problem in additive number theory, Math. Proc. Cambridge Philos. Soc. 103 (1) (1988), 2733.
3.Brüdern, J. and Kawada, K., The asymptotic formula in Waring's problem for one square and seventeen fifth powers, Monatsh. Math. 162 (4) (2011), 385407.
4.Davenport, H. and Heilbronn, H., On Waring's problem: two cubes and one square, Proc. London Math. Soc. (2) 43 (1) (1937), 73104.
5.Friedlander, J.B. and Wooley, T.D., On Waring's problem: Two squares and three biquadrates, Mathematika 60 (1) (2014), 153165.
6.Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, 5th ed. (The Clarendon Press, Oxford University Press, New York, 1979).
7.Hooley, C., On the representations of a number as the sum of two cubes, Math. Z. 82 (1963), 259266.
8.Hooley, C., On a new approach to various problems of Waring's type, Recent progress in analytic number theory, vol. 1 (Durham, 1979) (Academic Press, London-New York, 1981), 127191.
9.Hooley, C., On Waring's problem for two squares and three cubes Quart. J. Math. Oxford Ser. (2) 16 (1965), 289296.
10.Sinnadurai, J. St.-C. L., Representation of integers as sums of six cubes and one square, Quart. J. Math. Oxford Ser. 16 (2) (1965), 289296.
11.Stanley, G.K., The representation of a number as the sum of one square and a number of k-th powers, Proc. London Math. Soc. (2) 31 (1) (1930), 512553.
12.Vaughan, R.C., On the representation of numbers as sums of squares, cubes and fourth powers, PhD Thesis (London, 1969).
13.Vaughan, R.C., A ternary additive problem, Proc. London Math. Soc. (3) 41 (3) (1980), 516532.
14.Vaughan, R.C., On Waring's problem: One square and five cubes, Quart. J. Math. Oxford Ser. (2) 37 (145) (1986), 117127.
15.Vaughan, R.C., On Waring's problem for smaller exponents. II, Mathematika 33 (1) (1986), 622.
16.Vaughan, R.C., The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Mathematics, vol. 125 (Cambridge University Press, Cambridge, 1997).
17.Vaughan, R.C., On generating functions in additive number theory. I, Analytic number theory (Cambridge University Press, Cambridge, 2009), 436448.
18.Wooley, T.D., On Waring's problem: Some consequences of Golubeva's method, J. Lond. Math. Soc. (2) 88 (3) (2013), 699715.

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ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS

  • JÖRG BRÜDERN (a1)

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