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.
Citation
Badria Alsulmi. Vincent Pigno. Christopher Pinner. "Jacobi-type sums with an explicit evaluation modulo prime powers." Funct. Approx. Comment. Math. 56 (1) 49 - 60, March 2017. https://doi.org/10.7169/facm/1590
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