Organised as part of the "IHÉS Lectures", this Summer School aims to train PhD students, post-docs and young researchers on recent topics of Analytic Number Theory and to promote exchanges between young researchers of all nationalities.
Analytic number theory began with the first questions concerning the distribution of prime numbers. Since then, the subject has evolved in many directions; it has influenced and interacted with many areas of mathematics, by lending or borrowing ideas going from combinatorics to representation theory, and from modular forms to the deepest reaches of algebraic geometry.
The summer school will cover both classical and emerging topics of analytic number theory, with a focus on the properties of prime numbers:
(1) advanced sieve methods and their refinements, including approaches to gaps between primes and asymptotic sieve for primes;
(2) distribution of arithmetic functions in arithmetic progressions, especially in ranges beyond the direct reach of the Riemann Hypothesis;
(3) exponential sums over finite fields, and their analytic applications, with a focus on the formalism and uses of Frobenius trace functions;
(4) modular forms and associated L-functions, and other analytic aspects of the Langlands program, such as the behavior of torsion homology;
(5) additive combinatorics.