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A von Staudt-type result for \({\sum _{z\in \mathbb {Z}_n[i]} z^k }\)

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Abstract

In this paper we study the sum of powers of the Gaussian integers \(\mathbf {G}_k(n):=\sum _{a,b \in [1,n]} (a+b i)^k\). We give an explicit formula for \(\mathbf {G}_k(n) \pmod n\) in terms of the prime numbers \(p \equiv 3 \pmod 4\) with \(p \mid \mid n\) and \(p-1 \mid k\), similar to the well known one due to von Staudt for \(\sum _{i=1}^n i^k \pmod n\). We apply this result to study the set of integers \(n\) which divide \(\mathbf {G}_n(n)\) and compute its asymptotic density with six exact digits: \(0.971000\ldots \).

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Acknowledgments

The authors wish to thank Jonathan Sondow for his useful comments and remarks.

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Correspondence to Antonio M. Oller-Marcén.

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Communicated by J. Schoißengeier.

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Fortuny Ayuso, P., Grau, J.M. & Oller-Marcén, A.M. A von Staudt-type result for \({\sum _{z\in \mathbb {Z}_n[i]} z^k }\) . Monatsh Math 178, 345–359 (2015). https://doi.org/10.1007/s00605-015-0736-5

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  • DOI: https://doi.org/10.1007/s00605-015-0736-5

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