7,015 research outputs found
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
-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 -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 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)
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
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