Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum

Poorten, Alfred J. van der ; Williams, Kenneth S.

Canadian journal of mathematics, Tome 53 (2001) no. 2, pp. 434-448

Résumé

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

Détail

DOI : 10.4153/CJM-2001-018-8

Mots-clés : 11F20, 11E45, Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group

Bibliographie

[A] [A] van der Poorten, A. J. and Williams, K. S., Values of the Dedekind eta function at quadratic irrationalities. Canad. J. Math. 51 (1999), 176–224. Google Scholar