JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 2, Pages 209 - 259
(August 2004)
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EVALUATION
OF WEBER’S FUNCTIONS AT QUADRATIC
IRRATIONALITIES
Habib Muzaffar (Canada) and Kenneth S. Williams (Canada)
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Abstract: Let
denote the
Dedekind eta function. Let
be a positive-definite, primitive, integral,
binary quadratic form of discriminant
The value of
is determined for an arbitrary discriminant d.
This result generalizes the corresponding
result when d is fundamental, which was
obtained by van der Poorten and Williams [Canad.
J. Math. 51 (1999), 176-224. Corrigendum,
Canad. J. Math. 53 (2001), 434-448]. As a
consequence of our evaluation of
for
formulae are obtained for the moduli of Weber’s
functions
and
[Lehrbuch der Algebra, Vol. III, Chelsea
Publishing Co., New York, 1961, p. 114]. From
these formulae the values of
and
are determined for an arbitrary positive
integer m. |
Keywords and phrases: Dedekind eta function,
Weber’s functions, class invariants, binary
quadratic forms, Kronecker’s limit formula. |
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