Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics : CIRM Jean Morlet Chair Fall 2016 Part I. Bases of Analytic Number Theory 1. Prime Numbers 2. Arithmetic Functions 3. Dirichlet Series 4. Euler’s Gamma Function 5. Riemann’s Zeta Function 6. The Large Sieve 7. The Theorem of Vinogradov 8. The van der Corput Method. Part II. Interactions Between Arithmetics and Dynamics 9. A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems 10. Kloosterman Sums Disjointness and Equidistribution 11. Sarnak’s Conjecture what’s new 12. Sarnak’s Conjecture Implies the Chowla Conjecture Along a Subsequence 13. On the Logarithmic Probability that a Random Integral Ideal is A free 14. The Lagrange and Markov Spectra from the Dynamical Point of View 15. On the missing Log Factor 16. Chowla’s Conjecture: From the Liouville Function to the Moebius Function. Part III. Selected Topics in Dynamics 17. Weak Mixing for Infinite Measure Invertible Transformations 18. More on Tame Dynamical Systems 19. A Piecewise Rotation of the Circle IPR Maps and Their Connection with Translation Surfaces This book concentrates on the modern theory of dynamical systems and its interactions with number theory and combinatorics. The greater part begins with a course in analytic number theory and focuses on its links with ergodic theory presenting an exhaustive account of recent research on Sarnaks conjecture on Möbius disjointness. Selected topics involving more traditional connections between number theory and dynamics are also presented including equidistribution homogenous dynamics and Lagrange and Markov spectra. In addition some dynamical and number theoretical aspects of aperiodic order some algebraic systems and a recent development concerning tame systems are described