  This paper deals with a multiple version of zeta- and L-functions both in the
complex case and in the p-adic case: [I] Our motivation in the complex case is
to find suitable rigorous meaning of the values of multivariable multiple
zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs
turn to be entire on the whole space after taking the desingularization.
Further we show that the desingularized function is given by a suitable finite
linear combination of MZFs with some arguments shifted. It is also shown that
specific combinations of Bernoulli numbers attain the special values at their
non-positive integers of the desingularized ones. (b) Twisted MZFs can be
continued to entire functions and their special values at non-positive integer
points can be explicitly calculated. [II] Our work in the p-adic case is to
develop the study on analytic side of the Kubota-Leopoldt p-adic L-functions
(pLFs) into the multiple setting. We construct p-adic multiple L-functions
(pMLFs), multivariable versions of their pLFs, by using a specific p-adic
measure. We establish their various fundamental properties: (a) We establish
their intimate connection with the above complex MZFs by showing that the
special values of pMLFs at non-positive integers are expressed by the twisted
multiple Bernoulli numbers, the special values of the complex MZFs at
non-positive integers. (b) We extend Kummer congruence for Bernoulli numbers to
congruences for the twisted multiple Bernoulli numbers. (c) We extend the
vanishing property of the Kubota-Leopoldt pLFs with odd characters to our
pMLFs. (d) We establish their close relationship with the p-adic twisted
multiple polylogarithms (pTMPLs) by showing that the special values of pMLFs at
positive integers are described by those of pTMPLs at roots of unity, which
generalizes the previous result of Coleman in the single variable case.
