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

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

A note on the second moment of automorphic L-functions

We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet $L$-functions.Comment: 9 page

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

Gaps between zeros of the Riemann zeta-function

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist)

On the mean values of L-functions in orthogonal and symplectic families

Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices