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1. CJM 2011 (vol 63 pp. 1328)
On a Conjecture of Chowla and Milnor In this paper, we investigate a conjecture due to S. and P. Chowla and
its generalization by Milnor. These are related to the delicate
question of non-vanishing of $L$-functions associated to periodic
functions at integers greater than $1$. We report on some progress in
relation to these conjectures. In a different vein, we link them to a
conjecture of Zagier on multiple zeta values and also to linear
independence of polylogarithms.
Categories:11F20, 11F11 |
2. CJM 2001 (vol 53 pp. 434)
Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum 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 Categories:11F20, 11E45 |
3. CJM 1999 (vol 51 pp. 176)
Values of the Dedekind Eta Function at Quadratic Irrationalities 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 Categories:11F20, 11E45 |