Special values of partial zeta functions of real quadratic fields at nonpositive integers and Euler-Maclaurin formula

Abstract

We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field applying an asymptotic version of Euler-Maclaurin formula to the lattice cone associated to the ideal considered. The Euler-Maclaurin formula involved is obtained by applying the Todd series of differential operators to an integral of a small perturbation of thecone. The additive property of Todd series w.r.t. the cone decomposition enables us to express the partial zeta values in terms of the continued fraction of the reduced element of the ideal. The expression obtained uses the positive continued fraction which yields a virtual decomposition of the cone. We apply the expression to some indexed families of real quadratic fields satisfying certain condition on the shape of the continued fractions. The families considered include those appeared in \cite{J-L1} and \cite{J-L2} as well as the Richaud-Degert types. We show that the partial zeta values at a given nonpositive integer $-k$ in the family indexed by $n$ is a polynomial of $n$. Finally, we compute explicitly the polynomials producing the partial zeta values at $s=-k$ for small $k$ of some chosen families and compare these with some previously known results.