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Transactions of the American Mathematical Society

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

Author: David Kramer
Journal: Trans. Amer. Math. Soc. 299 (1987), 59-79
MSC: Primary 11R42; Secondary 11E32, 11M41
MathSciNet review: 869399
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Abstract: In 1976 Shintani gave a decomposition of the Dedekind zeta function, $ \zeta \kappa (s)$, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for $ \zeta \kappa ( - n),$, $ n = 0,\,1,\,2, \ldots $. Earlier, Zagier had studied the special case of $ \zeta (A,\,s)$, the narrow ideal class zeta function for a real quadratic field. He decomposes $ \zeta (A,\,s)$ into $ {\Sigma _A}{Z_Q}(s)$, where $ {Z_Q}(s)$ is given as a Dirichlet series associated to a binary quadratic form $ Q(x,\,y) = a{x^2} + bxy + c{y^2}$, and the summation is over a canonically given finite cycle of ``reduced'' quadratic forms associated to a narrow ideal class $ A$. He then obtains a formula for $ {Z_Q}( - n)$ as a rational function in the coefficients of the form $ Q$.

Since the denominator of $ \zeta (A,\, - n)$ is known not to depend on the class $ A$, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier's formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives

$\displaystyle 15120\zeta (A, - 2) = \sum\limits_A {\frac{{{b^5} - 10a{b^3}c + 3... ...}} + \frac{{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text{ }}$

while our result is

$\displaystyle 15120\zeta (A,\, - 2) = \frac{1} {2}\left( {\mathop \sum \limits_... ...mits_{\theta A} } \right)\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})$

, where $ \theta $ is the narrow ideal class consisting of principal ideals generated by elements of negative norm.

Starting with a representation of $ {Z_Q}(1 + n)$ due to Shanks and Zagier for $ n = 1,\,2,\,3, \ldots $ as a certain transcendental function of the coefficients of $ Q$, we also obtain the result that $ \zeta (A,\,1 + n)$ is given as the same sum of reduced quadratic forms as in the formula for $ \zeta (A,\, - n)$, times the appropriate ``gamma factor.'' This gives a new proof of the functional equation of $ \zeta (A,\,s)$ at integer values of $ s$, and suggests the possibility that one might be able to prove the functional equation for all $ s$ by finding some relation between $ {Z_Q}(s)$ and $ {Z_Q}(1 - s)$. So far we have not found such a relation.

References [Enhancements On Off] (What's this?)

  • [1] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR 0195803
  • [2] D. Shanks and D. Zagier, On the values of zeta functions of real quadratic fields at positive integers, in preparation. (Preprint available from Don Zagier, University of Maryland.)
  • [3] Don Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153–184. MR 0366877,
  • [4] D. Zagier, Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976) Soc. Math. France, Paris, 1977, pp. 135–151. Astérisque No. 41–42 (French). MR 0441925
  • [5] Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393–417. MR 0427231

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Article copyright: © Copyright 1987 American Mathematical Society

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