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Values of the Dedekind Eta Function at Quadratic Irrationalities

Published online by Cambridge University Press:  20 November 2018

Alfred van der Poorten  and
Kenneth S. Williams
Alfred van der Poorten
Affiliation:
Centre for Number Theory Research, School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW, Australia 2109 email: alf@mpce.mq.edu.au
Kenneth S. Williams
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6 email: williams@math.carleton.ca
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Abstract

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Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that

$${{b}^{2}}-4ac=d,a>0,\gcd (a,b,c)=1.$$
.

The value of $\left| \eta ((b+\sqrt{d})/\left. 2a) \right| \right.$ is determined explicitly, where $\eta \left( z \right)$ is Dedekind’s eta function

$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$

Keywords

11F2011E45Dedekind eta functionquadratic irrationalitiesbinary quadratic formsform class group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 1 , 01 February 1999 , pp. 176 - 224
Copyright
Copyright © Canadian Mathematical Society 1999

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