  Extending Gross's result, we prove that a certain factorizaton of measures
holds for all $p$ and any finite even Dirichlet character $\chi$ of any
conductor, rather than only for split $p$ and $\chi$ with conductor a power of
$p$. Using this generalization, we find lower bounds on the proportion of
imaginary quadratic fields $K$ for which (under certain assumptions on the
elliptic curve) a chosen quadratic twist of an elliptic curve $E$ over $K$ has
rank $1$. We also find lower and upper bounds for the proportion of quadratic
twists with rank $1$ when we vary $D$, the factor we twist by, under the
assumption that $\omega$ (the prime factor counting function) is sufficiently
close to a Gaussian distribution, as described by Erd\"os-Kac. We apply similar
methods to cubic twists, and then derive analogous lower bounds for the
proportion of imaginary quadratic fields for which a sextic twist has rank $1$.
Lastly, for elliptic curves over $\mathbb{Q}$ satisfying certain assumptions,
we find positive lower bounds on the proportion of quadratic twists (over
$\mathbb{Q}$) which have rank $0$ and rank $1$, which yields examples of
elliptic curves satisfying the weak Goldfeld conjecture.

Number Theory