  This work is a study of $p$-adic multiple zeta values at roots of unity
($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent
fundamental groupoid of $(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})/
\mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic
pro-unipotent harmonic actions. In this part IV we define and study $p$-adic
analogues of some elementary complex analytic functions which interpolate
multiple zeta values at roots of unity such as the multiple zeta functions. The
indices of $p$MZV$\mu_{N}$'s involve sequences of positive integers ; in this
IV-1, by considering an operation which we call localization (inverting certain
integration operators) in the pro-unipotent fundamental groupoid of
$\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$, and by using $p$-adic pro-unipotent
harmonic actions, we extend the definition of $p$MZV$\mu_{N}$'s to indices for
which these integers can be negative, and we study these generalized
$p$MZV$\mu_{N}$'s.
