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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
DOI: https://doi.org/10.1090/S0002-9939-1989-0977923-3
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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  • [1] W. L. Baily, Jr., Introductory lectures on automorphic forms, Princeton Univ. Press, 1973.
  • [2] Minking Eie, A zeta-function associated with zero ternary forms, Proc. Amer. Math. Soc. 94 (1985), 387-392. MR 787878 (86g:11022)
  • [3] Minking Eie and Chong-hsio Fang, On the residues and values of a zeta function at negative integers and negative half-integers, manuscript, 1987.
  • [4] I. M. Gelfand and G. E. Shilov, Generalized functions, vol. 1, 1964.
  • [5] David Kramer, On the values of integers of the Dedekind zeta function of a real quadratic field, Trans. Amer. Math. Soc. 299 (1987), 59-79. MR 869399 (88a:11123)
  • [6] A. Kurihara, On the values of nonpositive integers of Siegel's zeta functions of $ Q$-anisotropic quadratic forms with signature $ (1,n - 1)$, J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 28 (1981), 567-584. MR 656037 (84a:10021)
  • [7] Y. Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Math., vol. 812, Springer-Verlag, Berlin and New York, MR 584625 (82a:32034)
  • [8] -, A new compactification of the Siegel space and degeneration of abelian varieties. I, Math. Ann. 221 (1976), 97-141. MR 0480537 (58:697a)
  • [9] I. Satake, Special values of zeta functions associated with self-dual homogeneous cones, manuscript, 1981. MR 642867 (83h:10051)
  • [10] M. Sato, and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131-170. MR 0344230 (49:8969)
  • [11] T. Shintani, Zeta-functions associated with the vector of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 22 (1975), 25-65. MR 0384717 (52:5590)
  • [12] -, On evaluation of zeta functions of totally real algebraic number fields at nonpositive integers, J. Fac. Sci. Univ Tokyo 23 (1976), 393-417. MR 0427231 (55:266)
  • [13] C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), 527-606. MR 1503238
  • [14] -, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z. 43 (1938), 682-708. MR 1545742
  • [15] D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (19), 153-184. MR 0366877 (51:3123)
  • [16] -, Valeurs des fonctions zeta des corps quadratiques reèls aux entiers negatifs, J. Arithmétiques de Caen, Astérisque 41-42 (1977), 135-151. MR 0441925 (56:316)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0977923-3
Article copyright: © Copyright 1989 American Mathematical Society

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