zbMATH — the first resource for mathematics

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)].

11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
11B05 Density, gaps, topology
Full Text: DOI
[1] Beardon, AF, Sums of powers of integers, Am. Math. Month., 103, 201-213, (1996) · Zbl 0851.11012
[2] Carlitz, L.: The Staudt-Clausen theorem. Math. Mag. 34, 131-146 (1960-1961) · Zbl 0122.04702
[3] Grau, J.M., Oller-Marcén, A.M.: About the congruence \(∑ _{k=1}^n k^{f(n)} ≡ 0 (mod n)\). Preprint (2013). arXiv:1304.2678 · Zbl 0060.00919
[4] Kellner, B.C.: On the theorems of Von Staudt and Clausen. (In preparation) · Zbl 0445.05006
[5] Lengyel, T.: On divisibility of some power sums. Integers 7:A41, 6 (2007) · Zbl 1132.11302
[6] MacMillan, K; Sondow, J, Divisibility of power sums and the generalized Erdős-Moser equation, Elem. Math., 67, 182-186, (2012) · Zbl 1264.11025
[7] Moree, P, On a theorem of Carlitz-von staudt, C. R. Math. Rep. Acad. Sci. Can., 16, 166-170, (1994) · Zbl 0820.11002
[8] Sloane, N.J.A.: The on-line encyclopedia of integer sequences. https://oeis.org · Zbl 1274.11001
[9] Sondow, J., MacMillan, K.: Reducing the Erdős-Moser equation \(1^n+2^n+⋯ +k^n=(k+1)^n\) modulo \(k\) and \(k^2\). Preprint (2010). arXiv:1011.2154 · Zbl 1233.11038
[10] Sondow, J., MacMillan, K.: Reducing the Erdős-Moser equation \(1^n+2^n+⋯ +k^n=(k+1)^n\) modulo \(k\) and \(k^2\). Integers 11, A34, 8, (2011) · Zbl 1233.11038
[11] Staudt, KGC, Beweis eines lehrsatzes die bernoullischen zahlen betreffend, J. Reine Angew. Math, 21, 372-374, (1840) · ERAM 021.0672cj
[12] Dilcher, K.: Congruences for a class of alternating lacunary sums of binomial coeficients. J. Integer Seq. 10 Article (2007) · Zbl 1174.11002
[13] Hermite, Ch, Extrait d’une lettre a M. Borchardt, J. Reine Angew. Math., 81, 93-95, (1876) · JFM 07.0131.01
[14] Bachmann, P.: Niedere Zahlentheorie, Part 2, Teubner, Leipzig: parts 1 and 2 reprinted in one volume. Chelsea, New York, (1910) (1968) · Zbl 0820.11002
[15] Mathar, R.J.: Table of Dirichlet L-series and prime zeta modulo functions for small moduli (2010). arXiv:1008.2547 · Zbl 0851.11012
[16] Moser, L, On the Diophantine equation \(1^n+2^n+3^n+⋯ +(m-1)^n =m^n\), Scr. Math., 19, 84-88, (1953) · Zbl 0050.26604
[17] Butske, W; Jaje, LM; Mayernik, DR, On the equation \(∑ _{P ∣ N} \frac{1}{P}+\frac{1}{N}=1\), pseudoperfect numbers, and perfectly weighted graphs, Math. Comp, 69, 407-420, (2000) · Zbl 0934.11015
[18] Moree, P, Moser’s mathemagical work on the equation \(1^k+2^k+⋯ +(m-1)^k=m^k\), Rocky Mt. J. Math., 5, 1707-1737, (2013) · Zbl 1362.11045
[19] Guy, R.: Unsolved problems in number theory, 2nd edn. Springer, New York (2004) · Zbl 1058.11001
[20] Schultz, HJ, The sums of the kth powers of the first n integers amer, Math. Mon., 87, 478-481, (1980) · Zbl 0445.05006
[21] Boyer, CB, Pascal’s formula for the sums of powers of the integers, Scr. Math., 9, 237-244, (1943) · Zbl 0060.00919
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.