# Character sums with an explicit evaluation

Badria Alsulmi, Vincent Pigno and Christopher Pinner
From the journal Mathematica Slovaca

# Abstract

We show that for Dirichlet characters χ, χ1, …, χs mod pm the sum

x1=1pmxs=1pmχ1(x1)χs(xs)χ(A1x1++Asxs+Bx1w1xsws)

has a simple evaluation when m is sufficently large, for m ≥ 2 if

p2A1AsB(1w1ws).

(Communicated by Federico Pellarin)

### References

[1] Alsulmi, B.—Pigno, V.— Pinner, C.: Jacobi-type sums with an explicit evaluation modulo prime powers, (preprint at ), (to appear Funct. Approx. Comment. Math. (2017)).Search in Google Scholar

[2] Cochrane, T.: Exponential sums modulo prime powers, Acta Arith. 101 (2002), 131–149.Search in Google Scholar

[3] Cochrane, T.—Zheng, Z.: Pure and mixed exponential sums, Acta Arith. 91 (1999), 249–278.Search in Google Scholar

[4] Cochrane, T.—Zheng, Z.: A survey on pure and mixed exponential sums modulo prime powers. In: Number theory for the millennium I, Urbana, IL, 2000, pp. 273–300.Search in Google Scholar

[5] Long, M.—Pigno, V.—Pinner, C.: Evaluating prime power Gauss and Jacobi sums, arXiv:1410.6179 [math.NT], (to appear Tamkang J. Math.)Search in Google Scholar

[6] Pigno, V.—Pinner, C.: Binomial character sums modulo prime powers, J. Théor. Nombres Bordeaux 28 (2016), 39–53.Search in Google Scholar

[7] Pigno, V.—Pinner, C.—Sheppard, J.: Evaluating binomial character sums modulo powers of two, J. Math. Res. Appl. 35 (2015), 137–142.Search in Google Scholar

[8] Zhang, W.—Wang, T: A note on the Dirichlet characters of polynomials, Math. Slovaca 64 (2014), 301–310.Search in Google Scholar