Abstract view
Values of the Dedekind Eta Function at Quadratic Irrationalities


Published:19990201
Printed: Feb 1999
Alfred van der Poorten
Kenneth S. Williams
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
$$
\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1  e^{2\pi imz})
\qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).
$$