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

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- by Min King Eie
- Proc. Amer. Math. Soc.
**105**(1989), 273-280 - DOI: https://doi.org/10.1090/S0002-9939-1989-0977923-3
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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].## References

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## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**105**(1989), 273-280 - MSC: Primary 11R42; Secondary 11E12, 11R11
- DOI: https://doi.org/10.1090/S0002-9939-1989-0977923-3
- MathSciNet review: 977923