Limit Conjectures in Number Theory, Lehmer's Conjecture, Conjecture of Schinzel-Zassenhaus, Dynamical Zeta function of the Beta-shift: Part 4.-- Asymptotic Expansions of the Mahler Measures.Limit Equidistribution of Conjugates and Beta-conjugates (Bilu, Favre Rivera-Letellier), Theory of Erdös-Turan, Asymptotic Expansions and Polylogarithms : Poincaré, Condon. Dobrowolski type Inequalities and Minorations, Examples. Methods of Resolution.

This Summer School on the Theory of Motives and the Theory of Numbers, at the crossroad of several L functions, zeta functions, polyzeta functions and dynamical zeta functions will be held on the scientific campus of Le-Bourget-du-Lac, close to the Department of Mathematics (LAMA) of the University Savoie Mont Blanc, from June 18th till June 29th 2018. It is one of the thematic Schools of the CNRS whose main objective is to gather young researchers working in these domains, whose details can be found in the mini-courses. The topics of the mini-courses are closely related but often progress in parallel. The idea of this Summer School is to make them live in a transversal way. Several domains are concerned : in Geometry, singularity theory, model theory, o-minimality, motive theory, Igusa zeta functions, in Number Theory. p-adic and complex L functions on classical groups, Dirichlet series and zeta functions of one and several variables, in Dynamics of algebraic numbers, on Perron numbers, dynamical zeta functions of the beta-shift. The questions of rationality, of zeroes, of poles, of natural boundary, are often common to zeta, polyzeta, L functions