Special values of partial zeta functions of real quadratic fields at nonpositive integers and the Euler-Maclaurin formula (Q1941766)

The authors 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, applying the integral expression for the partial zeta value due to \textit{S. Garoufalidis} and \textit{J. E. Pommersheim} [J. Am. Math. Soc. 14, No. 1, 1--23 (2001; Zbl 1007.11067)]. A polynomial expression for the partial zeta values is obtained with variables given by the coefficient of the continued fraction. The partial zeta values are computed explicitly for \(s=0,-1,-2\), and the results compared with those obtained previously by \textit{D. Zagier} [Astèrisque 41--41, 135--151 (1977; Zbl 0359.12012)] and Garoufalidis and Pommersheim [loc. cit.]. The arithmetic significance of these values is discussed, and, as an application, class number one criteria are obtained for several families of real quadratic fields.