## 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.

### MSC:

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

### Keywords:

complete character sums

### Citations:

Zbl 0777.11030; Zbl 1030.11040; Zbl 1012.11077