Summer school in Paris 2021 : École d’été en théorie analytique des nombres

On a global scale, analytic number theory is currently undergoing spectacular and exciting developments. This is undoubtedly due to the creativity and dynamism of many young mathematicians. Matomäki and Radziwiłł revolutionised our understanding of the behaviour of multiplicative functions in short intervals. In subsequent collaborations with Tao, they established an average form of the Chowla conjecture, and extended their result to multiplicative functions twisted by additive characters. The study of the distribution of the Riemann zeta function on the critical line is a central topic. Harper, as well as Radziwiłł and his collaborators, have shown spectacular results concerning moments of the zeta functions and its maximum over short intervals. Koukoulopoulos and Maynard have recently fully resolved the Duffin-Schaeffer conjecture on rational approximations of real numbers. The dispersion method has had spectacular consequences on the distribution of arithmetic sequences in arithmetic progressions, including Zhang's result on bounded gaps between primes. We also intend to cover some problems of additive combinatorics type which involve tools coming from multiplicative number theory. The goal of this summer school is to transmit these new approaches, and to spark further discussions. It will allow beginning researchers to reinforce and broaden their knowledge of these new techniques.