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# Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum

Published:2001-04-01
Printed: Apr 2001
• Alfred J. van der Poorten
• Kenneth S. Williams
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## Abstract

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper [A] only the result $\pi_{K,d}(x)+\pi_{K^{-1},d}(x)=\frac{1}{h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)$ follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result $\pi_{K,d}(x)=\frac{1}{2h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)$ stated in Lemma~5.2. This necessitates changes in Sections~5 and 6 of [A]. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of [A], each and every occurrence of $o(s-1)$ should be replaced by $o(1)$. Sections~5 and 6 of [A] have been rewritten to incorporate the above mentioned correction and are given below. They should replace the original Sections~5 and 6 of [A].
 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)