1. CJM 2011 (vol 64 pp. 151)
 Miller, Steven J.; Wong, Siman

Moments of the Rank of Elliptic Curves
Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of HeathBrown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of lowlying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and SwinnertonDyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, lowlying zeros, quadratic twist, rank Categories:11G05, 11G40 

2. CJM 2005 (vol 57 pp. 267)
 Conrad, Keith

Partial Euler Products on the Critical Line
The initial version of the Birch and SwinnertonDyer conjecture
concerned asymptotics for partial Euler products for an elliptic curve
$L$function at $s = 1$. Goldfeld later proved that these asymptotics
imply the Riemann hypothesis for the $L$function and that the
constant in the asymptotics has an unexpected factor of $\sqrt{2}$.
We extend Goldfeld's theorem to an analysis of partial Euler products
for a typical $L$function along its critical line. The general
$\sqrt{2}$ phenomenon is related to second moments, while the
asymptotic behavior (over number fields) is proved to be equivalent to
a condition that in a precise sense seems much deeper than the Riemann
hypothesis. Over function fields, the Euler product asymptotics can
sometimes be proved unconditionally.
Keywords:Euler product, explicit formula, second moment Categories:11M41, 11S40 
