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A von Staudt-type result for \(\sum _{z\in \mathbb {Z}_n[i]} z^k\). (English) Zbl 1333.11018
After having recalled the basic literature about the sum of power of integers \[ S_k(n)=1^k+2^k+3^k+ \cdots +n^k \] [C. B. Boyer, Scripta Math. 9, 237–244 (1943; Zbl 0060.00919)]; [A. F. Beardon, Am. Math. Mon. 103, No. 3, 201–213 (1996; Zbl 0851.11012)] and about the sum modulo \(n\) ([L. Carlitz, Math. Mag. 34, 131–146 (1961; Zbl 0122.04702)]; [P. Moree, C. R. Math. Acad. Sci., Soc. R. Can. 16, No. 4, 166–170 (1994; Zbl 0820.11002)]; [T. Lengyel, Integers 7, No. 1, Paper A41, 6 p. (2007; Zbl 1132.11302)]; [K. MacMillan and J. Sondow, Elem. Math. 67, No. 4, 182–186 (2012; Zbl 1264.11025)]; J. Sondow and K. MacMillan [“Reducing the Erdős-Moser equation \(1^n + 2^n + \dots + k^n = (k + 1)^n\) modulo \(k\) and \(k^2\)”, arXiv:1011.2154], [J. Sondow and K. MacMillan, Integers 11, No. 6, 765–773, A34 (2011; Zbl 1233.11038)]), the authors clarify their aim to supply an analogue of the formula from Ch. von Staudt [J. Reine Angew. Math. 21, 372–374 (1840; ERAM 021.0672cj)]: \[ \sum_{i=1}^n i^k \pmod n \] in a Gaussian setting, i.e., based on the sum of powers of the Gaussian integers \(G_k(n):=\sum_{a,b \in [1,n]} (a+b i)^k\). The values of \(G_k(n) \pmod n \) for \(1 \leq k\) and \(n \leq 24\) are listed in a table.
The authors also compute, with six exact digits, the asymptotic density of the set of integers \(n\) dividing \(G_n(n)\) via PARI/GP and through the table of Dirichlet L-series and prime zeta modulo functions for small moduli provided by R. J. Mathar [“Table of Dirichlet \(L\)-series and prime zeta modulo functions for small moduli”, arXiv:1008.2547].
The authors report a computation, lasting over 24h, for the asymptotic density of the set of integers \(n\) such that \(n \mid G_n(n)\) and they remark that their value \(0.971000\dots\) is in contrast with the result \(0.5\) obtained in the classical integral setting \(n \mid S_n(n)\) by J. M. Grau, P. Moree and A. M. Oller-Marcén [“About the congruence \(\sum_{k=1}^n k^{f(n)} \equiv 0 \pmod n \)”, arXiv:1304.2678].
Beyond topical tools like the binomial theorem, the Chinese Remainder Theorem, the induction and the inclusion-exclusion principle, the proof employs some technical achievements by Ch. Hermite [Borchardt J. 81, 93–95 (1876; JFM 07.0131.01)], by K. Dilcher [J. Integer Seq. 10, No. 10, Article 07.10.1, 10 p. (2007; Zbl 1174.11002)] and by P. Bachmann [Niedere Zahlentheorie. Zweiter Teil: Additive Zahlentheorie. Leipzig: B. G. Teubner (1910; JFM 41.0221.10)], [Niedere Zahlentheorie. Teil 1. Teil 2: Additive Zahlentheorie. Reprint. Bronx, NY: Chelsea Publ. Co. (1968; Zbl 0253.10001)], involving sums of binomial coefficients.
The authors conclude wishing a paper’s further refinement by posing in a Gaussian contest even the Erdős-Moser equation investigated, e.g., by L. Moser [Scripta Math. 19, 84–88 (1953; Zbl 0050.26604)], W. Butske et al. [Math. Comput. 69, No. 229, 407–420 (2000; Zbl 0934.11015)], P. Moree [Rocky Mt. J. Math. 43, No. 5, 1707–1737 (2013; Zbl 1362.11045)] and illustrated by R. K. Guy [Unsolved problems in number theory. 3rd ed. New York, NY: Springer-Verlag (2004; Zbl 1058.11001)].

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
11B05 Density, gaps, topology
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