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Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum
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- Journal:
- Canadian Journal of Mathematics / Volume 53 / Issue 2 / 01 April 2001
- Published online by Cambridge University Press:
- 20 November 2018, pp. 434-448
- Print publication:
- 01 April 2001
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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 $\left[ \text{A} \right]$.
Values of the Dedekind Eta Function at Quadratic Irrationalities
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- Journal:
- Canadian Journal of Mathematics / Volume 51 / Issue 1 / 01 February 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 176-224
- Print publication:
- 01 February 1999
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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
$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$
