On the values at integers of the Dedekind zeta function of a real quadratic field
Trans. Amer. Math. Soc. 299 (1987), 59-79
Primary 11R42; Secondary 11E32, 11M41
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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.
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
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.)
Zagier, A Kronecker limit formula for real quadratic fields,
Math. Ann. 213 (1975), 153–184. MR 0366877
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
0441925 (56 #316)
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
- Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
- 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.)
- D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. MR 0366877 (51:3123)
- -, 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)
- 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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