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

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869399-4

MathSciNet review:
869399

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Abstract: In 1976 Shintani gave a decomposition of the Dedekind zeta function, , 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 , . Earlier, Zagier had studied the special case of , the narrow ideal class zeta function for a real quadratic field. He decomposes into , where is given as a Dirichlet series associated to a binary quadratic form , and the summation is over a canonically given finite cycle of ``reduced'' quadratic forms associated to a narrow ideal class . He then obtains a formula for as a rational function in the coefficients of the form .

Since the denominator of is known not to depend on the class , 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

Starting with a representation of due to Shanks and Zagier for as a certain transcendental function of the coefficients of , we also obtain the result that is given as the same sum of reduced quadratic forms as in the formula for , times the appropriate ``gamma factor.'' This gives a new proof of the functional equation of at integer values of , and suggests the possibility that one might be able to prove the functional equation for all by finding some relation between and . So far we have not found such a relation.

**[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**, https://doi.org/10.1007/BF01343950**[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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DOI:
https://doi.org/10.1090/S0002-9947-1987-0869399-4

Article copyright:
© Copyright 1987
American Mathematical Society