On the values at integers of the Dedekind zeta function of a real quadratic field. (English) Zbl 0606.12007
Let K be a real quadratic number field over the rationals. In 1977 D. Zagier [Astérisque 41/42, 135-151 (1977; Zbl 0359.12012)] has given a formula for the Dedekind zeta-function \(\zeta_ K(s,A)\) of the narrow ideal class A in K at nonpositive integers -n. He considers Dirichlet series \(Z_ Q(s)=\sum_{\lambda >0}a_{\lambda} \lambda^{-s}\) with \(\lambda =Q(p,q)\), the reduced binary quadratic form Q(p,q) associated to the ideal class A. It is possible to decompose \(\zeta_ K(s,A)\) into a finite sum of such Dirichlet series. Zagier then shows that the values of \(Z_ Q(-n)\) can be expressed in terms of rational functions in the coefficients of the form Q.
Since it is known that the denominator of \(\zeta_ K(-n,A)\) does not depend on A, whereas the coefficients of the reduced forms can be arbitrarily large, it is natural to ask whether the rational functions in Zagier’s formula might be replaced by polynomials. In the present paper such a result is obtained. Using a representation of \(Z_ Q(s)\) at positive integers due to Shanks and Zagier the author also gives a new proof of the functional equation for \(\zeta_ K(s,A)\) at integer values of s.
Since it is known that the denominator of \(\zeta_ K(-n,A)\) does not depend on A, whereas the coefficients of the reduced forms can be arbitrarily large, it is natural to ask whether the rational functions in Zagier’s formula might be replaced by polynomials. In the present paper such a result is obtained. Using a representation of \(Z_ Q(s)\) at positive integers due to Shanks and Zagier the author also gives a new proof of the functional equation for \(\zeta_ K(s,A)\) at integer values of s.
Reviewer: J.Hinz
MSC:
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11M35 | Hurwitz and Lerch zeta functions |
| 11E99 | Forms and linear algebraic groups |
Keywords:
values at integers; real quadratic number field; Dedekind zeta-function; narrow ideal class; reduced binary quadratic form; functional equationCitations:
Zbl 0359.12012References:
| [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. · Zbl 0145.04902 |
| [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. · Zbl 0283.12004 · doi: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). · Zbl 0359.12012 |
| [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. · Zbl 0349.12007 |
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