  It is widely accepted nowadays that polyzetas are connected by polynomial
relations. One way to obtain relations among polyzetas is to consider their
generating series and the relations among these generating series. This leads
to the indexation of the generating series of polylogarithms, recently
described in \cite{GHM22,BHN,CM}. But, in order to understand the bridge
between the extension of this "polylogarithmic calculus" and the world of
harmonic sums, a local theory of domains has to be done, preserving
quasi-shuffle identities, Taylor expansions and Hadamard products. In this
contribution, we present a sketched version of this theory. As an example of
generating series, one can consider the eulerian gamma function,
\begin{eqnarray*} \Gamma(1+z)=\exp\biggl(-\gamma z+\sum_{n\ge2}
\zeta(n)\dfrac{(-z)^n}{n}\biggr) {eqnarray*} and this may suggest to regularize
the divergent zeta value $\zeta(1)$, for the quasi-shuffle structure, as to be
Euler's $\gamma$ constant. In the same vein, in \cite{BHN}, we introduce a
family of eulerian functions, \begin{eqnarray*}
\Gamma_{y_k}(1+z)=\exp\biggl(\sum_{n\ge1}\zeta(kn)\dfrac{(-z^k)^n}{n}\biggr),
&\mbox{for}&k\ge2,y_k\in Y=\{y_n\}_{n\ge1}. {eqnarray*} This being done, in
this work, via their analytical aspects, we establish, on one side, their
existence and the fact that their inverses are entire. On the other side, using
the same symmetrization technique, we give their distributions of zeroes.
