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# Values of the Dedekind Eta Function at Quadratic Irrationalities

Published:1999-02-01
Printed: Feb 1999
• Alfred van der Poorten
• Kenneth S. Williams
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## Abstract

Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that $$b^2 - 4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.$$ The value of $\bigl|\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr|$ is determined explicitly, where $\eta(z)$ is Dedekind's eta function $$\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1 - e^{2\pi imz}) \qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).$$
 Keywords: Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group
 MSC Classifications: 11F20 - Dedekind eta function, Dedekind sums 11E45 - Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)

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