SPECIAL VALUES OF PARTIAL ZETA FUNCTIONS OF REAL QUADRATIC FIELDS AT NONPOSITIVE INTEGERS AND THE 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 in terms of the positive continued fraction of the reduced element defining the ideal. We apply the integral expression of the partial zeta value due to Garoufalidis-Pommersheim (2001) using the Euler-Maclaurin summation formula for a lattice cone associated to the ideal. From the additive property of Todd series w.r.t. the (virtual) cone decomposition arising from the positive continued fraction of the reduced element of the ideal, we obtain a polynomial expression of the partial zeta values with variables given by the coefficient of the continued fraction. We compute the partial zeta values explicitly for s = 0, -1, -2 and compare the result with earlier works of Zagier (1977) and Garoufalidis-Pommersheim (2001). Finally, we present a way to construct Yokoi-Byeon-Kim type class number one criterion for some families of real quadratic fields.