92 research outputs found
Moments and distribution of central L-values of quadratic twists of elliptic curves
We show that if one can compute a little more than a particular moment for
some family of L-functions, then one has upper bounds of the conjectured order
of magnitude for all smaller (positive, real) moments and a one-sided central
limit theorem holds. We illustrate our method for the family of quadratic
twists of an elliptic curve, obtaining sharp upper bounds for all moments below
the first. We also establish a one sided central limit theorem supporting a
conjecture of Keating and Snaith. Our work leads to a conjecture on the
distribution of the order of the Tate-Shafarevich group for rank zero quadratic
twists of an elliptic curve, and establishes the upper bound part of this
conjecture (assuming the Birch-Swinnerton-Dyer conjecture).Comment: 28 page
Large character sums: Burgess's theorem and zeros of -functions
We study the conjecture that for any primitive
Dirichlet character with , which is known to
be true if the Riemann Hypothesis holds for . We show that it holds
under the weaker assumption that `' of the zeros of up to
height lie on the critical line; and establish various other
consequences of having large character sums
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