In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{ Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{ F}_{ k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{ higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{ extended higher Herglotz function}. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by $\mathscr{ F}_{ k,N}(x)$. Radchenko and Zagier gave a beautiful relation between the integral $\displaystyle\int_{ 0}^{ 1}\frac{ \log(1+t^x)}{ 1+t}\, dt$ and $F(x)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $\mathscr{ F}_{ k,N}(x)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $\mathscr{ F}_{ k, N}(x)$ and some generalized Lambert series are also obtained along with other supplementary results.
Categories: math.NT math.CA