Zeros of L-functions and cancellation of modular coefficients along prime numbers

This thesis is divided into two main parts. In Chapter 1, we consider the average of modular coefficients over prime numbers, using the classical circle method. In Chapter 2 and 3, which correspond to the second part, we focus on Dirichlet series. In particular, in Chapter 2 we deal with the distribution of the zeros, giving an account of the main examples of Dirichlet series with infinitely many zeros in the region of absolute convergence. We prove the existence of zeros of this type for a generalized version of the Hurwitz zeta function. In Chapter 3, we consider this problem in the framework of the Selberg Class S of L-functions. We first give a general overview of the theory of S and its extension S]. Then, we focus on our main problem. Given a degree 1 function in S], we are interested in studying the analytic properties of its linear twists. We prove that the linear twists satisfy a functional equation of Hurwitz-Lerch type and we also give some results on the distribution of the zeros outside the critical strip