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Evaluation of the Dedekind Eta Function

Published online by Cambridge University Press:  20 November 2018

Robin Chapman  and
William Hart
Robin Chapman
Affiliation:
Department of Mathematical Sciences, University of Exeter, EX4 4QE, UK e-mail: [email protected]
William Hart
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. e-mail: [email protected]
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Abstract

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We extend the methods of Van der Poorten and Chapman for explicitly evaluating the Dedekind eta function at quadratic irrationalities. Via evaluation of Hecke $L$-series we obtain new evaluations at points in imaginary quadratic number fields with class numbers 3 and 4. Further, we overcome the limitations of the earlier methods and via modular equations provide explicit evaluations where the class number is 5 or 7.

Keywords

11G15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 1 , 01 March 2006 , pp. 21 - 35
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Abel, N. H., Recherches sur les fonctions elliptiques. J Reine Angew. Math. 2(1828), 380–382.Google Scholar
[2] Chapman, R. and van der Poorten, A. J., Binary quadratic forms and the eta function. In: Number Theory for the Millennium I, AK Peters, Natick MA, 2002, pp. 215–22.Google Scholar
[3] Cox, D. A., Primes of the Form x 2 + ny 2 : Fermat, Class Field Theory and Complex Multiplication. John Wiley & Sons, New York, 1989.Google Scholar
[4] Enge, A. and Morain, F., Comparing invariants for class fields of imaginary quadratic fields. In: Algorithmic Number Theory, Lectures Notes in Comput. Sci. 2369, Springer, Berlin, 2002, pp. 252–266.Google Scholar
[5] Fröhlich, A. and Taylor, M. J., Algebraic Number Theory. Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, Cambridge, 1993.Google Scholar
[6] Gee, A. and Stevenhagen, P., Generating class fields using Shimura reciprocity. In: Algorithmic Number Theory, Lecture Notes in Comput. Sci. 1423 Springer, Berlin, 1998, pp. 441–453.Google Scholar
[7] Hajir, F. and Villegas, F. Rodriquez, Explicit elliptic units. I. Duke Math. J. 90(1997), 495–521.Google Scholar
[8] Hart, William, Evaluation of the Dedekind eta function. Ph.D. Thesis, Macquarie University, Sydney, 2004.Google Scholar
[9] Hart, William, Schlaefli modular equations for generalized Weber functions. Ramanujan J., to appear.Google Scholar
[10] Huard, J. G., Kaplan, P., and Williams, K. S., The Chowla-Selberg formula for genera. Acta. Arith. 73(1995), no. 3, 271–301.Google Scholar
[11] Kaneko, M., A generalization of the Chowla-Selberg formula and the zeta functions of quadratic orders. Proc. Japan Acad. Ser. A Math. Sci. 66(1990), no. 7, 201–203.Google Scholar
[12] Nakkajima, Y. and Taguchi, Y., A generalization of the Chowla-Selberg formula. J. Reine Angew. Math. 419(1991), 119–124.Google Scholar
[13] PARI-GP. See http://pari.math.u-bordeaux.fr. Google Scholar
[14] van der Poorten, A. J. and Williams, K. S., Values of the Dedekind eta function at quadratic irrationalities. Canad. J. Math. 51(1999), no. 1, 176–224.Google Scholar
[15] Selbert, A. and Chowla, S., On Epstein's zeta-function. J. Reine Agnew. Math. 227(1967), 86–110.Google Scholar
[16] Weber, H., Lehrbuch der Algebra. vol. 3, 3rd Edition. Chelsea, NY, 1979.Google Scholar
[17] Zhang, N. Y. and Williams, K. S., On the Epstein zeta function. Tamkang J. Math. 26(1995), no. 2, 165–176.Google Scholar
[18] Zucker, I. J. and Robertson, M. M., Exact values for some two-dimensional lattice sums. J. Phys. A 8(1975), 874–881.Google Scholar
[19] Zucker, I. J. and Robertson, M. M., Some properties of Dirichlet L-serie. J. Phys. A 9(1976), 1207–1214.Google Scholar
[20] Zucker, I. J. and Robertson, M. M., A systematic approach to the evaluation of Σ(m, n)≠(0,0)(am2 + bmn + cn2)–s . J. Phys. A 9(1976), 1215–1225.Google Scholar
[21] Zucker, I. J. and Robertson, M. M., Further aspects of the evaluation of Σ(m, n)≠(0,0)(am2 + bmn + cn2)–s . Math. Proc. Cambridge. Philos. Soc. 95(1984), 5–13.Google Scholar