Skip to main content Accessibility help
×
Home

On Waring's problem for cubes

  • Jörg Brüdern (a1)

Extract

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shape

where (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

Copyright

References

Hide All
[1]Brüdern, J.. Sums of four cubes. Monatsh. Math. 107 (1989), 179188.
[2]Brüdern, J.. On Waring's problem for fifth powers and some related topics. Proc. London Math. Soc. (3) 61 (1990), 457479.
[3]Brüdern, J.. Pairs of diagonal cubic forms. Proc. London Math. Soc. (3) 61 (1990), 273343.
[4]Davenport, H.. On Waring's problem for cubes. Acta Math. 71 (1939), 123143.
[5]Gallagher, P. X.. The large sieve. Mathematika 14 (1967), 1420.
[6]Gallagher, P. X.. A large sieve density estimate near σ = 1. Invent. Math. 11 (1970), 329339.
[7]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. 5th edition (Oxford University Press, 1979).
[8]Hooley, C.. On the Barban–Davenport–Halberstam theorem III. J. London Math. Soc. (2) 10 (1975), 249256.
[9]Hooley, C.. On Waring's problem. Acta Math. 157 (1986), 4797.
[10]Hua, L.-K.. Some results in the additive prime number theory. Quart. J. Math. Oxford Ser. (2) 9 (1938), 6880.
[11]Roth, F. K.. On Waring's problem for cubes. Proc. London Math. Soc. (2) 53 (1951), 268279.
[12]Vaughan, R. C.. The Hardy–Littlewood Method (Cambridge University Press, 1981).
[13]Vaughan, R. C.. Some remarks on Weyl sums. In Topics in Classical Number Theory. Colloq. Math. Soc. János Bolyai no. 34 (North-Holland, 1981), pp. 15851602.
[14]Vaughan, R. C.. On Waring's problem for cubes. J. Reine Angew. Math. 365 (1986), 121170.
[15]Vaughan, R. C.. A new iterative method in Waring's problem. Acta Math. 162 (1989), 171.
[16]Vaughan, R. C.. On Waring's problem for cubes II. J. London Math. Soc. (2) 39 (1989), 205218.

On Waring's problem for cubes

  • Jörg Brüdern (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed