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Extending Gross's result, we prove that a certain factorizaton of measuresholds for all $p$ and any finite even Dirichlet character $\chi$ of anyconductor, rather than only for split $p$ and $\chi$ with conductor a power of$p$. Using this generalization, we find lower bounds on the proportion ofimaginary quadratic fields $K$ for which (under certain assumptions on theelliptic curve) a chosen quadratic twist of an elliptic curve $E$ over $K$ hasrank $1$. We also find lower and upper bounds for the proportion of quadratictwists with rank $1$ when we vary $D$, the factor we twist by, under theassumption that $\omega$ (the prime factor counting function) is sufficientlyclose to a Gaussian distribution, as described by Erd\"os-Kac. We apply similarmethods to cubic twists, and then derive analogous lower bounds for theproportion 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 ofelliptic curves satisfying the weak Goldfeld conjecture.

Factorization of measures and applications to the weak Goldfeld conjecture