Pré-soutenance : Sur quelques identités entre les valeurs spéciales des fonctions L

The main goal this talk is to study the relations among some special zeta values. It contains three main parts. In the first part, we study the zeta values in Tate algebras which is introduced by Pellarin in 2012. We give an affirmative answer to a conjecture of Pellarin about identities for these zeta values. We also suggest a conjecture about the coefficients of a special several variables polynomials which is closely related to zeta values in Tate algebras. In the second part, we generalize Speyer's results (2017) in the context of rank one Drinfeld modules which is related to the zeta values of Goss. Lastly, we study the multiple zeta values which is introduced by Thakur in 2004. We prove a conjecture of Lara Rodriguez and Thakur which gives a full list of depth 2 zeta-like of a specific bounded weight. We also give a similar result about determining completely all zeta-like of weight of that bound.