Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Note on Heath-Brown's estimate
for Heilbronn's exponential sum


Author: Hong Bing Yu
Journal: Proc. Amer. Math. Soc. 127 (1999), 1995-1998
MSC (1991): Primary 11L03
Published electronically: March 17, 1999
MathSciNet review: 1487349
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that $S_h(a)=\sum^p_{n=1}e(\frac{an^{hp}}{p^2})\ll (h,p-1)p^{11/12}$, which generalizes Heath-Brown's estimate for Heilbronn's exponential sum $S_1(a)$. We also give a simple proof of a crucial lemma in Heath-Brown's work.


References [Enhancements On Off] (What's this?)

  • 1. D. R. Heath-Brown, An estimate for Heilbronn’s exponential sum, Analytic number theory, Vol. 2 (Allerton Park, IL, 1995) Progr. Math., vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 451–463. MR 1409372
  • 2. R. W. K. Odoni, Trigonometric sums of Heilbronn’s type, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 3, 389–396. MR 803598, 10.1017/S0305004100063593
  • 3. S. A. Stepanov, The number of points of a hyperelliptic curve over a finite prime field, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1171–1181 (Russian). MR 0252400

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Additional Information

Hong Bing Yu
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, The People’s Republic of China
Email: yuhb@ustc.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04776-0
Received by editor(s): August 13, 1997
Received by editor(s) in revised form: October 23, 1997
Published electronically: March 17, 1999
Additional Notes: Supported by the National Science Foundation of China
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society