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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the values at negative half-integers of the Dedekind zeta function of a real quadratic field


Author: Min King Eie
Journal: Proc. Amer. Math. Soc. 105 (1989), 273-280
MSC: Primary 11R42; Secondary 11E12, 11R11
MathSciNet review: 977923
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Abstract: The zeta function $ \zeta (A,s)$ associated with a narrow ideal class $ A$ for a real quadratic field can be decomposed into $ \sum\nolimits_Q {{Z_Q}(s)} $, where $ {Z_Q}(s)$ is a Dirichlet series associated with a quadratic form $ Q(x,y) = a{x^2} + bxy + c{y^2}$, and the summation is over finite reduced quadratic forms associated to the narrow ideal class $ A$. The values of $ {Z_Q}(s)$ at nonpositive integers were obtained by Zagier [16] and Shintani [12] via different methods. In this paper, we shall obtain the values of $ {Z_Q}(s)$ at negative half-integers $ s = - 1/2, - 3/2, \ldots , - m + 1/2, \ldots $. The values of $ {Z_Q}(s)$ at nonpositive integers were also obtained by our method, and our results are consistent with those given in [16].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0977923-3
PII: S 0002-9939(1989)0977923-3
Article copyright: © Copyright 1989 American Mathematical Society