  Multizeta values are numbers appearing in many different contexts.
Unfortunately, their arithmetics remains mostly out of reach. In this article,
we define a functional analogue of the algebra of multizetas values, namely the
algebra of multitangent functions, which are 1- periodic functions defined by a
process formally similar to multizeta values. We introduce here the fundamental
notions of reduction into monotangent functions, projection onto multitangent
functions and that of trifactorization, giving a way of writing a multitangent
function in terms of Hurwitz multizeta functions. This explains why the
multitangent algebra is a functional analogue of the algebra of multizeta
values. We then discuss the most important algebraic and analytic properties of
these functions and their consequences on multizeta values, as well as their
regularization in the divergent case. Each property of multitangents has a
pendant on the side of multizeta values. This allows us to propose new
conjectures, which have been checked up to the weight 18.
