Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T03:49:02.043Z Has data issue: false hasContentIssue false

On Waring's problem for smaller exponents. II

Published online by Cambridge University Press:  26 February 2010

R. C. Vaughan
Affiliation:
Department of Mathematics, Imperial College London, Queen's Gate, London. SW7 2BZ
Get access

Extract

Let k denote a fixed natural number with k > 2, let ℛs(n) denote the number of representations of n as the sum of sk-th powers of natural numbers, and let

with

denote the corresponding singular series.

Type
Research Article
Copyright
Copyright © University College London 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Davenport, H.. On Waring's problem for fourth powers. Ann. Math., 40 (1939), 731747.CrossRefGoogle Scholar
2.van der Corput, J. C.. Une inequality relative au nombre des diviseurs. Ned. Akad. Wet. Proc., 42 (1939), 547553.Google Scholar
3.Greaves, G.. On the representation of a number as a sum of two fourth powers. Math. Z., 94 (1966), 223234.CrossRefGoogle Scholar
4.Halberstam, H. and Richert, H.-E.. Sieve Methods (Academic Press, London, 1974).Google Scholar
5.Hall, R. R. and Tenenbaum, G.. The average order of Hooley's Δr-functions II. Compositio Math. To appear.Google Scholar
6.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”; IV Math. Z., 12 (1922), 161188.Google Scholar
7.Hooley, C.. On a new technique and its applications to the theory of numbers. Proc. London Math. Soc., (3), 38 (1979), 115151.CrossRefGoogle Scholar
8.Hooley, C.. On another sieve method and the numbers that are representable as the sum of two h-th powers. Proc. London Math. Soc., (3), 43 (1981), 73109.CrossRefGoogle Scholar
9.Hua, L. K.. On Waring's problem. Quart. J. Math., 9 (1938), 199202.CrossRefGoogle Scholar
10.Hua, L. K.. Additive Theory of Prime Numbers, A.M.S. translations of mathematical monographs, Vol. 13 (Providence, 1965).Google Scholar
11.Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge University Press, London, 1981).Google Scholar
12.Vaughan, R. C.. Some remarks on Weyl sums. Coll. Math. Soc. János Bolyai, Budapest, 1981. 34: Topics in Classical Number Theory (Elsevier North Holland, 1984), 15851602.Google Scholar
13.Vaughan, R. C.. On Waring's problem for cubes. J. für reine und ang. Math. To appear.Google Scholar
14.Wirsing, E.. Das asymptotisch Verhalten von Summen über multiplikative Funktionen. Math. Ann., 143 (1961), 75102.CrossRefGoogle Scholar