Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-18T23:10:07.565Z Has data issue: false hasContentIssue false
  • English
  • Français

INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

Part of: Algebraic number theory: global fields Zeta and $L$-functions: analytic theory Sequences and sets Series expansions

Published online by Cambridge University Press:  26 February 2015

WILLIAM DUKE  and
HA NAM NGUYEN
WILLIAM DUKE*
Affiliation:
Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, CA 90095-1555, USA email wdduke@ucla.edu
HA NAM NGUYEN
Affiliation:
Department of Mathematics, California State University, 18111 Nordhoff Street, Northridge, CA 91330, USA email ha.nguyen@csun.edu
*
✉wdduke@ucla.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

Keywords

infinite productscyclotomic polynomialsnatural boundaryMahler functionsStern diatomic sequence

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions
Secondary: 11B85: Automata sequences 11R18: Cyclotomic extensions 30B30: Boundary behavior of power series, over-convergence
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 400 - 411
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Allouche, J.-P. and Shallit, J., ‘The ubiquitous Prouhet–Thue–Morse sequence’, in: Sequences and their Applications (Singapore, 1998), Springer Series in Discrete Mathematics and Theoretical Computer Science (Springer, London, 1999), 1–16.Google Scholar
Allouche, J.-P., Mendès France, M. and Peyrière, J., ‘Automatic Dirichlet series’, J. Number Theory 81 (2000), 359–373.Google Scholar
Apostol, T., ‘Resultants of cyclotomic polynomials’, Proc. Amer. Math. Soc. 24 (1970), 457–462.CrossRefGoogle Scholar
de Bruijn, N. G., ‘On Mahler’s partition problem’, Ned. Akad. Wet. Proc. 51 (1948), 659–669; Indagationes Math. 10 (1948), 210–220.Google Scholar
Bundschuh, P., ‘Algebraic independence of infinite products and their derivatives’, in: Number Theory and Related Fields, Springer Proceedings in Mathematics and Statistics, 43 (Springer, New York, 2013), 143–156.Google Scholar
Coons, M., ‘The transcendence of series related to Stern’s diatomic sequence’, Int. J. Number Theory 6(1) (2010), 211–217.CrossRefGoogle Scholar
Coons, M., ‘(Non)automaticity of number theoretic functions’, J. Théor. Nombres Bordeaux 22 (2010), 339–352.CrossRefGoogle Scholar
Coquet, J., ‘A summation formula related to the binary digits’, Invent. Math. 73(1) (1983), 107–115.CrossRefGoogle Scholar
Drmota, M. and Stoll, Th., ‘Newman’s phenomenon for generalized Thue–Morse sequences’, Discrete Math. 308(7) (2008), 1191–1208.CrossRefGoogle Scholar
Dumas, P. and Flajolet, P., ‘Asymptotique des récurrences mahlériennes: le cas cyclotomique’, J. Théor. Nombres Bordeaux 8(1) (1996), 1–30.Google Scholar
Kurshan, R. P. and Odlyzko, A. M., ‘Values of cyclotomic polynomials at roots of unity’, Math. Scand. 49(1) (1981), 15–35.Google Scholar
Lehmer, D. H., Mahler, K. and van der Poorten, A. J., ‘Integers with digits 0 or 1’, Math. Comp. 46(174) (1986), 683–689.Google Scholar
Mahler, K., ‘Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen’, Math. Ann. 101 (1929), 342–366.Google Scholar
Mahler, K., ‘Über die Taylorcoeffizienten rationaler Funktionen’, in: Akad. Amsterdam, 38 (1935), 51–60.Google Scholar
Mahler, K., ‘On a special functional equation’, J. Lond. Math. Soc. (2) 15 (1940), 115–123.Google Scholar
Mahler, K., ‘On two analytic functions’, Acta Arith. 49(1) (1987), 15–20.Google Scholar
Nagell, T., Introduction to Number Theory (Wiley, New York, 1951).Google Scholar
Newman, D. J., ‘On the number of binary digits in a multiple of three’, Proc. Amer. Math. Soc. 21 (1969), 719–721.CrossRefGoogle Scholar
Nguyen, H. N., Dirichlet series associated to $B$-multiplicative arithmetic functions, Master’s Thesis, California State University, 2014.Google Scholar
Pellarin, F., ‘An introduction to Mahler’s method for transcendence and algebraic independence’, Preprint, 2005, arXiv:1005.1216v2.Google Scholar
Pólya, G. and Szegő, G., Problems and Theorems in Analysis. II. Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry (Springer, Berlin, 1998).Google Scholar
Remmert, R., Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, 172 (Springer, New York, 1998), Translated from the German by Leslie Kay.Google Scholar
Stanley, R. P. and Wilf, H. S., Refining the Stern Diatomic sequence, Preprint available at http://www-math.mit.edu/∼rstan/papers/stern.pdf.Google Scholar
Stern, M. A., ‘Über eine zahlentheoretische Funktion’, J. reine angew. Math. 55 (1858), 193–220.Google Scholar