Estimates for exponential sums
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- by Robert A. Smith PDF
- Proc. Amer. Math. Soc. 79 (1980), 365-368 Request permission
Abstract:
If f is a polynomial over Z of degree $n + 1$ with $n \geqslant 1$, then for each integer $q \geqslant 1,|{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)| \leqslant {q^{1/2}}(D,q){d_n}(q)$, 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
- Jing Run Chen, On Professor Hua’s estimate of exponential sums, Sci. Sinica 20 (1977), no. 6, 711–719. MR 480375
- 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), no. 1-2, 14–27. MR 1544448, DOI 10.1007/BF01378332
- Loo-keng Hua, On an exponential sum, J. Chinese Math. Soc. 2 (1940), 301–312. MR 4259
- L. K. Hua, Additive theory of prime numbers, Translations of Mathematical Monographs, Vol. 13, American Mathematical Society, Providence, R.I., 1965. MR 0194404
- Gyula Sándor, Uber die Anzahl der Lösungen einer Kongruenz, Acta Math. 87 (1952), 13–16 (German). MR 47679, DOI 10.1007/BF02392280
- André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. MR 27006, DOI 10.1073/pnas.34.5.204
Additional Information
- © Copyright 1980 American Mathematical Society
- 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