Explicit approaches to L-functions and primes

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The Riemann Hypothesis is one of the most sought after conjectures in mathematics. Its application to the distribution of prime numbers is fundamental. The relation between zeros of the Riemann zeta function and primes is at the origin of the field of analytic number theory. The philosophy of this program consists in optimizing and creating analytic number theory tools in order to produce completely explicit results which are numerically relevant. Prominent examples of this strategy can be found in the work of Helfgott on the Goldbach conjecture. Fully descriptive results can also then be directly applied to other fields of mathematics like combinatorics or cryptography. The Riemann Hypothesis and its generalized versions assert that all complex non-trivial zeros of the Riemann zeta function (and other more general families of L-functions) sit along the vertical line passing through ½. Our understanding of primes relies essentially on how far left from the vertical line passing through 1 the zeros are located and how many of them there are. I plan to prove new zero-density results in order to investigate questions about primes between consecutive powers. Unlike the zeta function, L-functions have ``low-lying" zeros (with small imaginary part), and possibly one close to 1 (referred to as "exceptional"). These zeros actually play a significant role in understanding primes (in arithmetic progressions, in number fields). A useful information is the fact that the possible exceptional zero has a repulsion effect on other zeros. I propose to investigate the location and density of zeros. This program includes enlarging zero-free regions and having stronger repulsion phenomenon for Dirichlet and Hecke L-functions. I also want to exhibit scarcity of zeros near the 1-line. I would apply these results to explore several prime number theorems. Together with various smoothing arguments, sieve bounds, and numerical computations, I aim to improve previous explicit estimates for various finite sums and product over prime numbers. Many of the tools developed to study zeros of Dirichlet L-functions and primes in arithmetic progressions can be generalized to Hecke L-functions and to the context of Chebotarev density theorem. In particular I would explore the size of the least prime and of error terms in the Chebotarev density theorem. I am also interested in applications to the Lang-Trotter conjecture and to bounds for primality testing.