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Estimates for exponential sums


Author: Robert A. Smith
Journal: Proc. Amer. Math. Soc. 79 (1980), 365-368
MSC: Primary 10G10
DOI: https://doi.org/10.1090/S0002-9939-1980-0567973-5
MathSciNet review: 567973
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Abstract: If f is a polynomial over Z of degree $ n + 1$ with $ n \geqslant 1$, then for each integer $ q \geqslant 1,\vert{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)\vert \leqslant {q^{1/2}}(D,q){d_n}(q)$, provided the discriminant D of the derivative of f does not vanish identically, where $ {d_n}(q)$ is the number of representations of q as a product of n factors.


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

  • [1] Jing-Run Chen, On Professor Hua's estimate of exponential sums, Sci. Sinica 20 (1977), 711-719. MR 0480375 (58:542)
  • [2] G. H. Hardy and J. E. Littlewood, Some problems of ``Partitio numerorum": II. Proof that every large number is the sum of at most 21 biquadrates, Math. Z. 9 (1921), 14-27. MR 1544448
  • [3] L. K. Hua, On an exponential sum, J. Chinese Math. Soc. 2 (1940), 301-312. MR 0004259 (2:347h)
  • [4] -, Additive theory of prime numbers, Transl. Math. Mono., vol. 13, Amer. Math. Soc., Providence, R. I., 1965. MR 0194404 (33:2614)
  • [5] G. Sándor, Über die Anzahl der Lösungen einer Kongruenz, Acta Math. 87 (1952), 13-17. MR 0047679 (13:913d)
  • [6] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. MR 0027006 (10:234e)

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DOI: https://doi.org/10.1090/S0002-9939-1980-0567973-5
Article copyright: © Copyright 1980 American Mathematical Society

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