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), 5979
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, , 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 while our result is , where is the narrow ideal class consisting of principal ideals generated by elements of negative norm. 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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708693994
PII:
S 00029947(1987)08693994
Article copyright:
© Copyright 1987 American Mathematical Society
