Kronecker second limit formula for real quadratic fields

Abstract

In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier, which connects Zagier's zeta function to the zeta function of real quadratic fields, we express the values of the zeta function of narrow ideal classes in real quadratic fields at natural arguments in terms of an analytic function which we call the \emph{higher Herglotz-Zagier-Novikov function} and denote it by $\mathscr{F}_k(x; \alpha, \beta)$. This function plays a central role in our study. The function $\mathscr{F}_k(x; \alpha, \beta)$ possesses elegant properties, for example, we prove that it satisfies the two, three and six-term functional equations. As a result of our Kronecker limit formula and functional equations, we provide another expression for the combinations of zeta values. Finally, we interpret our Kronecker ``limit" formula in terms of cohomological relations and establish a connection between $\mathscr{F}_k(x; \alpha, \beta)$ and a generalized Dedekind-eta function.