Bounds for certain character sums. (English) Zbl 1082.11051

The authors study the complete character sums \(S(\chi, f, p^m)=\sum_{x\bmod p^m}\chi(f(x))\) where \(f\) is a polynomial with integer coefficients and \(\chi\) is a primitive Dirichlet character modulo \(p^m\). They first provide examples (similar to the one given by [D. Ismoilov, Acta Math. Sin., New Ser. 9, No.1, 90–99 (1993; Zbl 0777.11030)] in the case of exponential sums) of polynomials \(f_0\) of arbitrary degree \(d\) and constant term arbitrary \(b\) for which \(S(\chi, f_0, p^m)= \chi(b)p^{m(1-1/d)}\) (\(d\neq 2,4\) if \(p=2\)). Then they extend Weyl and Estermann bounds for Kloosterman sum by taking \(f=a/x+bx\) and prove that \(| S(\chi,f,p^m)| \leq 2 p^{m/2}\). The novelty here is in case \(m\geq 2\) where they exploits a lemma of [T. Cochrane and Z. Zheng, Asian J. Math. 4, No. 4, 757–774 (2000; Zbl 1030.11040)] while the study of the examples uses a similar lemma taken from [T. Cochrane, Z. Zheng, Proc. Am. Math. Soc. 129, No. 9, 2505–2516 (2001; Zbl 1012.11077)] this time in the context of complete exponential sums. Note equation (2) is faulty (take \(q_1=1\) for instance), but one can indeed reduce \(S(\chi,f,q)\) to \(S(\chi,f,q_1)\) in a similar fashion.


11L40 Estimates on character sums
11L05 Gauss and Kloosterman sums; generalizations
11L10 Jacobsthal and Brewer sums; other complete character sums