## Functiones et Approximatio Commentarii Mathematici

### Jacobi-type sums with an explicit evaluation modulo prime powers

#### Abstract

We show that for Dirichlet characters $\chi_1,\ldots ,\chi_s$ mod $p^m$ the sum $$\mathop{\sum_{x_1=1}^{p^m} \dots \sum_{x_s=1}^{p^m}}_{ A_1x_1^{k_1}+\dots+ A_sx_s^{k_s}\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_s(x_s),$$ has a simple evaluation when $m$ is sufficiently large.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 56, Number 1 (2017), 49-60.

Dates
First available in Project Euclid: 27 January 2017

https://projecteuclid.org/euclid.facm/1485486017

Digital Object Identifier
doi:10.7169/facm/1590

Mathematical Reviews number (MathSciNet)
MR3629010

Zentralblatt MATH identifier
06864145

#### Citation

Alsulmi, Badria; Pigno, Vincent; Pinner, Christopher. Jacobi-type sums with an explicit evaluation modulo prime powers. Funct. Approx. Comment. Math. 56 (2017), no. 1, 49--60. doi:10.7169/facm/1590. https://projecteuclid.org/euclid.facm/1485486017

#### References

• B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi Sums, Canadian Math. Soc. series of monographs and advanced texts, vol. 21, Wiley, New York 1998.
• T. Cochrane and Z. Zheng, Pure and mixed exponential sums, Acta Arith. 91 (1999), no. 3, 249–278.
• M. Long, V. Pigno, and C. Pinner, Evaluating Prime Power Gauss and Jacobi Sums, arXiv:1410.6179 [math.NT].
• I. Niven, H.S. Zuckerman, and H. L. Montgomery, An Introduction to The Theory of Numbers, 5th edition, John Wiley & Sons, Inc. 1991.
• V. Pigno and C. Pinner, Binomial Character Sums Modulo Prime Powers, J. Théor. Nombres Bordeaux 28 (2016), no. 1, 39–53.
• W. Zhang and T. Wang, A note on the Dirichlet characters of polynomials, Math. Slovaca 64 (2014), no. 2, 301–310.
• W. Zhang and W. Yao, A note on the Dirichlet characters of polynomials, Acta Arith. 115 (2004), no. 3, 225–229.
• W. Zhang and Z. Xu, On the Dirichlet characters of polynomials in several variables, Acta Arith. 121 (2006), no. 2, 117–124.