Gaps between zeros of the derivative of the Riemann \xi-function

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive proportion of gaps are greater than 1.18 times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than 0.7203 (1.5, respectively).Comment: 15 page

Central values of derivatives of Dirichlet L-functions

Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method we show that L^{(k)}(1/2,chi) is not equal to zero for almost all the characters chi in C(q,+) when k and q are large. Here, L^{(k)}(s,chi) is the k-th derivative of of the Dirichlet L-function L(s,chi).Comment: submitted for publicatio

Twists of automorphic L-functions at the central point

We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.\chi,s), and from that derive an upper bound for the average analytic rank of the studied L-functions

Large gaps between consecutive zeros of the Riemann zeta-function. II

Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing

A note on the gaps between consecutive zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and infinitely often they differ by at least 2.69 times the average spacing.Comment: 7 pages. Submitted for publicatio