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On Hua’s estimates for exponential sums. (English) Zbl 0621.10024

Let f(x)\(\in {\mathbb{Z}}[X]\), \(e_ q(t)=\exp (2\pi i/q)\) and \(S(q,f)=\sum _{0\leq x<q}e_ q(f(x))\). Suppose henceforth that \(q=p^{\ell}\) is a prime power. Hua Lookeng [cf. Enzyklop√§die Math. Wiss. Bd. I, 2, H. 13, Art. 29 (Teubner 1959; Zbl 0083.036), § 13, S. 41] mentions the estimate \[ | S(p^{\ell},f)| = O[p^{\ell [1-(1/M+1)]}]\quad, \] where M denotes the maximum multiplicity of the roots of the congruence \(p^{-t} f'(x)\equiv 0 mod p,\) where \(p^{-t} f'(X)\) is primitive mod p. Here, it is given an explicit form of the type \[ | S(p^{\ell},f)| \leq mk p^{t/(M+1)} p^{\ell [1-(1/M+1)]}\quad, \] where m is the sum of the multiplicities and \(k=\deg f\). Some examples provide a comparison with the recent estimate of J. H. Loxton and R. C. Vaughan [Can. Math. Bull. 28, 440-454 (1985; Zbl 0575.10033)].

MSC:

11L40 Estimates on character sums
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[1] DOI: 10.1073/pnas.34.5.204 · Zbl 0032.26102
[2] Loxton, Canad. Math. Bull. 28 pp 440– (1985) · Zbl 0575.10033
[3] DOI: 10.1215/S0012-7094-57-02406-7 · Zbl 0088.03901
[4] Hua, Additiv Primzahltheorie pp 2– (1959)
[5] Chalk, Canad. Math. Bull. 30 pp 257– (1987) · Zbl 0595.10029
[6] DOI: 10.1112/jlms/s2-26.1.15 · Zbl 0474.10030
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