Comparative prime number theory and L-functions

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Number theory is one of the oldest branches of pure mathematics, and one of the largest. It has several important applications notably to data encryption and cryptography, numerical analysis, and signal transmission. A central question in number theory is to understand the distribution of prime numbers. In 1859 Riemann established a revolutionary formula for the counting function of the primes in terms of the zeros of the Riemann zeta function (which is the analytic continuation of an infinite series that is defined only on half of the complex plane). In his groundbreaking paper he proposed the conjecture that all the nontrivial zeros of the Riemann zeta function have real part equal 1/2. The Riemann Hypothesis is one of the greatest unsolved problems in mathematics and one of the seven millennium prize problems. The Riemann zeta function belongs to a large class of functions called L-functions. The analytic properties and the zeros of L-functions are connected to various arithmetic, algebraic and geometric objects such as prime numbers, elliptic curves, modular forms, and automorphic representations. My research is concerned with the study of L-functions and their applications, notably to questions related to the distribution of primes numbers in arithmetic progressions. In particular, my research explores the connection between the zeros of Dirichlet L-functions and comparative prime number theory, which is the study of the discrepancies of distributions when we compare the number of primes in different arithmetic progressions. In addition, my research will also investigate the distribution of values of L-functions and their moments in the critical strip.