Character sums, L- functions and applications

The theory of character sums and L- functions started with Dirichlet in his proof of the equidistribution of prime numbers in arithmetic progressions. Since then, it turned out not only that these concepts have been extended in great generality, but also that they can be conveniently used in many problems of number theory and cryptography. For instance, bounds on short char- acter sums give strong estimates on the least quadratic residue modulo a prime and the least primitive root. Essentially, the bests currently known bounds go back to Burgesss breakthrough result in 1957. Some refinements and generalizations of his result brought into the picture the importance of the notion of multiplicative energy. The developing area of additive combinatorics provides new tools to study it and opens new interesting possible horizons. Besides, the related notion of GCD sums was intensively studied recently due to the connection with large values of the Riemann zeta function and more generally large values of L- functions. Originally GCD sums had also interesting applications in metric Diophantine approximation. Although character sums and L- functions are intensively investigated fields of mathematics, a number of important problems are still open. Our research project focus on several questions of analytic and combinatorial nature. Meanwhile, we plan to develop further applications of character sums in number theoretical problems. One of the goals of the project is to investigate several minimization problems involving GCD sums and the multiplicative energy which arised in our recent joint work with de la Bretèche and Tenenbaum. We discovered several natural applications to character sums as well as non-vanishing of modular forms. We aim to develop new techniques which are appropriate for solving such questions and to formalize precisely the counterpart optimization problems in other settings: quadratic characters, Hecke cuspforms, finite fields, sums over generalized arithmetic progressions/Bohr sets. Another related topic of this project is the study of generalized Sidon sets. One aim is to obtain new deterministic and probabilistic constructions of large sub-Sidon sets. Additionally, one aims to study different kind of GCD sums using methods of additive combinatorics and look for applications in the field of metric Diophantine approximation. Finally, another focus of the project concerns analytic properties of L- functions and automorphic forms. Several problems such as large values of L- functions and moments of L- functions will be considered using previous techniques and potentially new efficient methods. We also plan to investigate zeros of Fekete polynomials and other connected problems using probabilistic techniques. 1

During the project, I made several progress on questions related to character sums and L- functions which were mentioned in the proposal as well as new questions which arised later. To briefly summarize for a non specialist, these functions encode very precisely the distribution of the prime numbers . In a work with S. Louboutin, we solved a question of E. Elma about the average value of these functions over certain subgroups of Dirichlet characters. Our method uses both recent results on the distribution of character sums and techniques from uniform distribution. This question has important applications to estimate the so called class number, a quantity which measure the failure of unique factorization in a context generalizing the classical integers. On another topic I wrote two papers with C. Aistleitner and D. El-Baz concerning local statistics of sequences (pair correlation and minimal gaps). These questions come originally in quantum mechanics, where the spacings of energy levels of integrable systems were studied. The ultimate goal is to show that some specific deterministic sequences involving primes or squares behave essentially as a random sequence. In our work, we used methods coming from analytic number theory such as L-functions to attack these apparently not related problems. We formulate a number of problems and conjectures which already motivated further research in the literature. I also worked on problems related to the greatest common divisor, a well-known quantity studied in elementary schools. In a paper with de la Bretèche and Tenenbaum, we solved a question concerning sequences minimizing a specific quantity defined using the greatest common divisors. It turned out to have applications in other fields of mathematics such as modular forms.