## 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. Enzyklopädie 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

### Citations:

Zbl 0083.036; Zbl 0575.10033
Full Text:

### References:

 [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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