The group of squarefree integers. (English) Zbl 1352.11003
Summary: We investigate the consequences of the elementary observation that the squarefree numbers form a group under the operation \(\frac{\text{lcm}}{\gcd}\). In particular, we discuss the characters on this group, one of which is the Möbius function, as well as the finite subgroups \(D(k)\) formed from the divisors of a given squarefree integer \(k\). We show further how a convolution, naturally based on this operation, leads to the factorization of various arithmetical matrices and the evaluation of the eigenvalues. We discuss briefly the associated \(L\)-functions. Finally, we generalize the operation to other subsets of \(\mathbb{N}\).
MSC:
| 11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 20K99 | Abelian groups |
References:
| [1] | Aistleitner, C.; Berkes, I.; Seip, K., GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc. (JEMS) (2014), (in press) |
| [2] | Apostol, T. M., Introduction to Analytic Number Theory (1976), Springer · Zbl 0335.10001 |
| [3] | Borwein, P.; Choi, S.; Coons, M., Completely multiplicative functions taking values in \(\{- 1, 1 \}\), Trans. Amer. Math. Soc., 362, 6279-6291 (2010) · Zbl 1222.11111 |
| [4] | Conrad, K., The origin of representation theory, Enseign. Math., 44, 361-392 (1998) · Zbl 0966.01018 |
| [5] | Dyer, T.; Harman, G., Sums involving common divisors, J. Lond. Math. Soc., 34, 1-11 (1986) · Zbl 0602.10041 |
| [6] | Gál, I. S., A theorem concerning Diophantine approximations, Nieuw Arch. Wiskd., 23, 13-38 (1949) · Zbl 0031.25601 |
| [7] | Grabowski, A., On squarefree numbers, Formaliz. Math., 21, 153-162 (2013) · Zbl 1298.11008 |
| [8] | Haukkanen, P.; Ilmonen, P.; Nalli, A.; Sillanpää, J., On unitary analogs of GCD reciprocal LCM matrices, Linear Multilinear Algebra, 58, 599-616 (2010) · Zbl 1231.11032 |
| [9] | Hawkins, T., The origins of the theory of group characters, Arch. Hist. Exact Sci., 7, 142-170 (1971) · Zbl 0217.29903 |
| [10] | Marsden, J. E., Elementary Classical Analysis (1974), Freeman and Company · Zbl 0285.26005 |
| [11] | Nathanson, M. B., Elementary Methods in Number Theory (2000), Springer · Zbl 0953.11002 |
| [12] | Perelli, A.; Zannier, U., An extremal property of the Möbius function, Arch. Math., 53, 20-29 (1989) · Zbl 0683.10036 |
| [13] | Rotman, J. J., Advanced Modern Algebra, Grad. Stud. Math. (2002), American Mathematical Society · Zbl 0997.00001 |
| [14] | Rozanov, Y. A., Probability Theory: A Concise Course (1977), Dover Publications · Zbl 0397.60001 |
| [15] | Tóth, L., A survey of Gcd-sum functions, J. Integer Seq., 13 (2010), Article 10.8.1, 23 pp · Zbl 1206.11118 |
| [16] | Wintner, A., The Theory of Means in Arithmetical Semi-Groups (1944), Waverly Press: Waverly Press Baltimore, MD · Zbl 0060.10512 |
| [17] | Wintner, A., Diophantine approximations and Hilbert’s space, Amer. J. Math., 66, 564-578 (1944) · Zbl 0061.24902 |
| [18] | Wirsing, E., Das Asymptotische Verhalten von Summen über multiplikative Funktionen II, Acta Math. Acad. Sci. Hungar., 18, 411-467 (1967) · Zbl 0165.05901 |
| [19] | Zhang, F., Matrix Analysis (2011), Springer |
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