On the Values at Integers of the Dedekind Zeta Function of a Real Quadratic Field

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 \[ 15120\zeta (A, - 2) = \sum \limits _A {\frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{a^3}}}} + \frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text { }}\] while our result is \[ 15120\zeta (A, - 2) = \frac {1} {2}\left ( {\sum \limits _A - \sum \limits _{\theta A} } \right ) (60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})\], 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.