Equivalence criteria for the two-term functional equations for Herglotz-Zagier functions

Abstract

For any integer $a$ and non-negative integer $b$, we define a Herglotz--Zagier (HZ) type function $F_{a,b}(x)$ by an absolutely convergent series involving the Digamma function $\psi(x)$. For each such $F_{a,b}(x)$, we associate an integer weight. In the literature, Ramanujan, Guinand, Zagier, Vlasenko-Zagier have derived two-term functional equations for some HZ type functions of positive weights. In this paper, we study a class of HZ type function associated with negative weights, and obtain their two-term functional equations. Parallelly, we associate an integer weight to the Kronecker limit type formula for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$. We establish that any two-term functional equation for HZ type function is equivalent to a Kronecker limit type formula of $\Theta(r,s,t,x)$, preserving weight. As a consequence, we derive new Kronecker limit type formulas and obtain a new special value of the Mordell--Tornheim zeta function $\zeta_{\textup{MT}}(r,s,t)$. We also obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to show that the Mordell--Tornheim zeta function lies centrally between many known modular relations.