Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-07T23:58:26.029Z Has data issue: false hasContentIssue false
  • English
  • Français

The Estimation of Complete Exponential Sums

Published online by Cambridge University Press:  20 November 2018

J. H. Loxton  and
R. C. Vaughan
J. H. Loxton
Affiliation:
University of New South WalesSydney, Australia
R. C. Vaughan
Affiliation:
Imperial CollegeLondon, England
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.

This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

Keywords

10G1012B05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 440 - 454
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Anderson, D.R. and Stiffler, J.T., Lower bounds for the maximum modulus of certain classes of trigonometric sums, Duke Math. J., 30 (1963), pp. 171–176.Google Scholar
2. Chen, J.-R., On the representation of natural numbers as a sum of terms of the form [x(x + 1)… (x + n -1] / n!, Acta Math. Sinica, 9 (1959), pp. 264–270.Google Scholar
3. Chen, J.-R., On Professor Hua's estimate of exponential sums, Sci. Sinica, 20 (1977), pp. 711—719.Google Scholar
4. Davenport, H. and Heilbronn, H., On an exponential sum, Proc. London Math. Soc, (2), 41 (1936), pp. 449–453.Google Scholar
5. Davenport, H. and Heilbronn, H., On Waring's problem: Two cubes and one square, Proc. London Math. Soc, (2), 43 (1937), pp. 73–104.Google Scholar
6. Gauss, K.F., Summatio quarundam serierum singularium, Comment. Soc. Reg. Sci. Gottingen Recentiores 1 (1808/11) (see also, Werke, Band II, Kônigl. Ges. Wiss. Göttingen, 1863,1876).Google Scholar
7. Hardy, G.H. and Littlewood, J.E., Some problems of “Partitio Numerorum“; I: A new solution of Waring's problem, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Math.-phys. Klasse, (1920), pp. 33–54.Google Scholar
8. Hardy, G.H. and Littlewood, J.E., Some problems of “Partitio Numerorum” : VI: Further researches in Waring's Problem, Math. Z., 23 (1925), pp. 1–37.Google Scholar
9. Hua, L.-K., On an exponential sum, J. Chinese Math. Soc, 2 (1940), pp. 301–312 (see also, Sur une somme exponentielle, C. R. Acad. Sci. Paris, 210 (1940), pp. 520–523).Google Scholar
10. Hua, L.-K., On exponential sums, Sci. Record (Peking) (N.S.), 1 (1957), pp. 1–4.Google Scholar
11. Loxton, J.H. and Smith, R.A., On Hua's estimate for exponential sums, J. London Math. Soc. (2), 26 (1982), pp. 15–20.Google Scholar
12. Mordell, L.J., On a sum analogous to a Gauss's sum, Quart. J. of Math., 3 (1932), pp. 161—167.Google Scholar
13. Nechaev, V.I., On the representation of natural numbers as a sum of terms of the form (x(x + 1)… (x + n - 1))/n!, Izv. Akad. Nauk SSSR Ser. Mat., 17 (1953), pp. 485–498.Google Scholar
14. Nechaev, V.I., An estimate of a complete rational trigonometric sum, Mat. Zametki, 17 (1975), pp. 839–849 (English translation, Math. Notes, 17 (1975), pp. 504–511).Google Scholar
15. Schmidt, W.M., Equations over finite fields. An elementary approach, Lecture Notes in Mathematics, 536, Springer-Verlag, Berlin, 1976.Google Scholar
16. Stechkin, S.B., An estimate of Gaussian sums, Mat. Zametki, 17 (1975), pp. 579–588 (English translation, Math. Notes, 17 (1975), pp. 342–349).Google Scholar
17. Stechkin, S.B., Estimate of a complete rational trigonometric sum, Proc. Steklov Inst., 143 (1977), pp. 188–220 (English translation, A.M.S., 1980, Issue 1, pp. 201-220).Google Scholar
18. Vaughan, R.C., The Hardy-Littlewood method, C.U.P., 1981, Cambridge.Google Scholar
19. Vinogradov, I.M., The method of trigonometric sums in the theory of numbers, Proc. Steklov Inst., 23(1947), pp. 1–111.Google Scholar
20. Weil, A., On some exponential sums, Proc. Nat. Acad. Sci., U.S.A., 34 (1948), pp. 204–207.Google Scholar
21. Weyl, H., On the equidistribution of numbers mod. one, Math. Ann., 77 (1916), pp. 313—352.Google Scholar