Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of $p$-adic multiple $L$-functions at arbitrary integers (Various aspects of multiple zeta values)

Abstract

We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give 'desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as a finite sum of ordinary multiple zeta-functions of the same type. As applications, we explicitly compute special values of desingularized double zeta-functions of Euler-Zagier type. We also extend our previous results concerning a relationship between p-adic multiple L-functions and p-adic multiple star polylogarithms to more general indices with arbitrary (not necessarily all positive) integers.