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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] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [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] D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. MR 0366877 (51:3123)
  • [4] -, Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, J. Arithmétiques de Caen, Astérisque 41-42 (1977), 135-151. MR 0441925 (56:316)
  • [5] Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields on non-positive integers, J. Fac. Sci. Univ. Tokyo 23 (1976), 393-417. MR 0427231 (55:266)

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Article copyright: © Copyright 1987 American Mathematical Society

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