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

$p$-adic multiple zeta values at roots of unity and $p$-adic  pro-unipotent harmonic actions - IV-1 : $p$-adic multiple zeta values at  roots of unity extended to sequences of integers of any sign