On the irreducibility of the Dirichlet polynomial of an alternating group
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Abstract:
Given a finite group $G$ the Dirichlet polynomial of $G$ is \[ P_{G}(s)=\sum _{H\leq G} \frac {\mu _G(H)}{|G:H|^s},\] where $\mu _G$ is the Möbius function of the subgroup lattice of $G$. This object is a member of the factorial domain of finite Dirichlet series. In this paper we prove that if $G$ is an alternating group of degree $k$ and $k\leq 4.2\cdot 10^{16}$ or $k\geq (e^{e^{15}}+2)^3$, then $P_G(s)$ is irreducible. Moreover, assuming the Riemman Hypothesis, we prove that $P_G(s)$ is irreducible in the remaining cases.References
- Nigel Boston, A probabilistic generalization of the Riemann zeta function, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 155–162. MR 1399336
- Yuan-You Fu-Rui Cheng, Explicit estimate on primes between consecutive cubes, Rocky Mountain J. Math. 40 (2010), no. 1, 117–153. MR 2607111, DOI 10.1216/RMJ-2010-40-1-117
- E. Detomi and A. Lucchini, Crowns and factorization of the probabilistic zeta function of a finite group, J. Algebra 265 (2003), no. 2, 651–668. MR 1987022, DOI 10.1016/S0021-8693(03)00275-8
- E. Damian and A. Lucchini, Finite groups with $p$-multiplicative probabilistic zeta function, Comm. Algebra 35 (2007), no. 11, 3451–3472. MR 2362665, DOI 10.1080/00927870701509313
- Erika Damian, Andrea Lucchini, and Fiorenza Morini, Some properties of the probabilistic zeta function on finite simple groups, Pacific J. Math. 215 (2004), no. 1, 3–14. MR 2060491, DOI 10.2140/pjm.2004.215.3
- P. Dusart. Estimates of some function over primes without R.H. 2010.
- P. Erdös. A theorem of Sylvester and Schur. J. London Math. Soc., S1-9 no. 4, 282–288, 1934.
- The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12.
- P. Hall. The Eulerian functions of a group. Quart. J. Math., 7:134–151, 1936.
- T. Hawkes, I. M. Isaacs, and M. Özaydin, On the Möbius function of a finite group, Rocky Mountain J. Math. 19 (1989), no. 4, 1003–1034. MR 1039540, DOI 10.1216/RMJ-1989-19-4-1003
- Camille Jordan, Traité des substitutions et des équations algébriques, Librairie Scientifique et Technique Albert Blanchard, Paris, 1957 (French). Nouveau tirage. MR 0091260
- Camille Jordan, Sur la limite de transitivité des groupes non alternés, Bull. Soc. Math. France 1 (1872/73), 40–71 (French). MR 1503635
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- D. H. Lehmer, On a problem of Störmer, Illinois J. Math. 8 (1964), 57–79. MR 158849
- L. J. Lander and T. R. Parkin, On first appearance of prime differences, Math. Comp. 21 (1967), 483–488. MR 230677, DOI 10.1090/S0025-5718-1967-0230677-4
- Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc. 86 (1990), no. 432, iv+151. MR 1016353, DOI 10.1090/memo/0432
- Avinoam Mann, Positively finitely generated groups, Forum Math. 8 (1996), no. 4, 429–459. MR 1393323, DOI 10.1515/form.1996.8.429
- Marilena Massa, The probabilistic zeta function of the alternating group $\textrm {Alt}(p+1)$, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), no. 3, 581–591 (English, with English and Italian summaries). MR 2351530
- Jitsuro Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952), 177–181. MR 50615
- Bertil Nyman and Thomas R. Nicely, New prime gaps between $10^{15}$ and $5\times 10^{16}$, J. Integer Seq. 6 (2003), no. 3, Article 03.3.1, 6. MR 1997838
- M. Patassini. On the irreducibility of the Dirichlet polynomial of a simple group of Lie type. Accepted by Israel J. Math.
- M. Patassini. Recognizing the non-Frattini abelian chief factors from the Probabilistic Zeta function of a finite group. In preparation, 2010.
- Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II, Math. Comp. 30 (1976), no. 134, 337–360. MR 457374, DOI 10.1090/S0025-5718-1976-0457374-X
- Paz Jiménez-Seral, Coefficients of the probabilistic function of a monolithic group, Glasg. Math. J. 50 (2008), no. 1, 75–81. MR 2381734, DOI 10.1017/S0017089507004053
- Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260, DOI 10.1017/CBO9780511805967
Additional Information
- Massimiliano Patassini
- Affiliation: Dipartimento di Matematica, Università di Padova, Via Trieste, 63 - 35121 Padova, Italia
- Email: frapmass@gmail.com
- Received by editor(s): May 12, 2011
- Received by editor(s) in revised form: June 11, 2011, and June 19, 2011
- Published electronically: April 2, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4041-4062
- MSC (2010): Primary 11M41; Secondary 11N05, 20D06, 20E28
- DOI: https://doi.org/10.1090/S0002-9947-2013-05655-3
- MathSciNet review: 3055688