Questions around the nontrivial zeros of the Riemann zeta-function. Computations and classifications

We study the sequence of nontrivial zeros of the Riemann zeta-function with respect to sequences of zeros of other related functions, namely, the Hurwitz zeta-function and the derivative of Riemann's zeta-function. Finally, we investigate connections of the nontrivial zeros with the periodic zeta-function. On the basis of computation we derive several classifications of the nontrivial zeros of the Riemann zeta-function and stateproblems which mightbe ofinterestfor abetter understanding of the distribution of those zeros

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