Abstract :

The Kronecker limit formulas are concerned with the constant term in the Laurent series expansion of certain Dirichlet series at $s=1$. The Kronecker limit formulas involve various special functions such as the \emph{Herglotz function} $F(x)$, and the functions $J(x)$ and $T(x)$. These functions have appeared in the works of Herglotz, Muzaffar and Willliams, and Zagier. Recently, Radchenko and Zagier extensively studied the properties of the Herglotz function such as its special values, its connection to Stark's conjecture, etc. After providing an overview of the literature in this research area, we will discuss the arithmetic properties of a Herglotz-type function that appears in a Kronecker limit formula derived by Novikov. For example, we will present the two- and three-term functional equations satisfied by it along with its special values. If time permits, we will also discuss about the second limit formula of Kronecker. This talk is based on the joint papers with Professor YoungJu Choie.