Weighted statistics of L-functions

This thesis consists of four chapters. Chapter 1 and 2 are a general introduction to the Riemann zeta function, with a special focus on the theory of moments; no original results are contained here. In Chapter 3 we study a weighted value distribution of the Riemann zeta function on the critical line. More specifically, assuming the Riemann Hypothesis, we investigate the distribution of $log|zeta(1/2+it)|$ with respect to various tilted measures, proving several weighted analogues of Selberg's central limit theorem. Moreover we prove unconditionally the analogue results in the corresponding random matrix theory setting. These contents first appeared in [54,55]. Chapter 4 is devoted to a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. First, we study this problem for the Riemann zeta function, both unconditionally and assuming the ratio conjecture. Then we generalize these ideas for specific families of $L$-functions with different symmetry types; in particular we consider a symplectic and an orthogonal family of $L$-functions and, under the relevant ratio conjecture, we study the weighted one-level density of non-trivial zeros of these $L$-functions.