Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-13T17:52:34.489Z Has data issue: false hasContentIssue false
  • English
  • Français

MINOR ARC MOMENTS OF WEYL SUMS

Published online by Cambridge University Press:  02 August 2012

M. P. HARVEY
M. P. HARVEY*
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom e-mail: michael.harvey@rhul.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

Keywords

11L1511P55
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 97 - 113
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Birch, B. J., Davenport, H. and Lewis, D. J., The addition of norm forms, Mathematika 9 (1962), 75–82.Google Scholar
2.Boklan, K. D., A reduction technique in Waring's problem. I, Acta Arith. 65 (2) (1993), 147–161.Google Scholar
3.Boklan, K. D., The asymptotic formula in Waring's problem, Mathematika 41 (2) (1994), 329–347.Google Scholar
4.Harvey, M. P., On certain cubic forms in seven variables, Math. Proc. Cambridge Philos. Soc. 149 (1) (2010), 21–47.Google Scholar
5.Hua, L. K., Some results in the additive prime-number theory, Q. J. Math. Oxford 9 (1938), 68–80.CrossRefGoogle Scholar
6.Hua, L. K., Additive theory of prime numbers, in Translations of mathematical monographs, vol. 13 (Amer. Math. Soc., Providence, RI, 1965).Google Scholar
7.Isaacs, I., Character theory of finite groups (AMS Chelsea Publishing, Providence, RI, 2006).Google Scholar
8.Iwaniec, H. and Kowalski, E., Analytic number theory (American Mathematical Society Colloquium Publications, Providence, RI, 2004).Google Scholar
9.Kawada, K. and Wooley, T. D., On the Waring–Goldbach problem for fourth and fifth powers, Proc. London Math. Soc. 83 (1) (2001), 1–50.CrossRefGoogle Scholar
10.Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I. Classical theory, in Cambridge studies in advanced mathematics, vol. 97 (Cambridge University Press, Cambridge, UK, 2007).Google Scholar
11.Moroz, B. Z., Analytic arithmetic in algebraic number fields, Lecture Notes in Mathematics, vol. 1205 (Springer-Verlag, Berlin, 1986).Google Scholar
12.Vaughan, R. C., On Waring's problem for cubes, J. Reine Angew. Math. 365 (1986), 122–170.Google Scholar
13.Vaughan, R. C., On Waring's problem for smaller exponents. II, Mathematika 33 (1) (1986), 6–22.Google Scholar
14.Vaughan, R. C., A new iterative method in Waring's problem, Acta Math. 162 (1–2) (1989), 1–71.CrossRefGoogle Scholar
15.Vaughan, R. C., The Hardy–Littlewood method, second ed. (Cambridge University Press, Cambridge, UK, 1997).Google Scholar