Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions

Abstract

The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F\left(x\right)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $\Theta (z,x)$, we show that the theories of these functions are inextricably woven. We obtain a three-term functional equation for $\Theta (z,x)$ as well as decompose it in terms of the Herglotz-Hurwitz function $\Phi (z,x)$. This decomposition can be conceived as a two-term functional equation for $\Phi (z,x)$. Through this result, we are not only able to get Zagier's identity relating $F\left(x\right)$ with $F(1/x)$ but also a two-term functional equation for Ishibashi's generalization of $F\left(x\right)$, namely, ${\Phi }_k\left(x\right)$, which has been sought after for over twenty years. We further generalize $\Theta (z,x)$ by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Choie and Kumar, were known so far. The first one among the two involves the integral $J\left(x\right)$ whose special values have received a lot of attention, more recently, in the work of Muzzaffar and Williams, and in that of Radchenko and Zagier. Analytic continuation of our generalization of $\Theta (z,x)$ is also accomplished which allows us to obtain transformations between certain double series and Herglotz-type integrals or their explicit evaluations.