Klagsbrun–Mazur–Rubin average-rank conjecture for quadratic twists

Let $K$ be a number field, let $E$ be an elliptic curve over $K$, and let $C(K,X)$ denote the set of quadratic characters of $K$ of norm less than $X$. Write $E^\chi$ for the quadratic twist of $E$ by $\chi$, $\operatorname{rk}(E^\chi)$ for its analytic rank, and let

$$\kappa=\prod_v\kappa_v,$$

where the local factors $\kappa_v$ are defined in the source. Klagsbrun–Mazur–Rubin conjecture. The average analytic rank of the quadratic twists of $E$ satisfies

$$\lim_{X\rightarrow\infty}\frac{\displaystyle\sum_{\chi\in C(K,X)}\operatorname{rk}(E^\chi)}{|C(K,X)|}=\frac{1+(-1)^{\operatorname{rk}(E)}\kappa}{2}.$$

The paper's parity theorem gives the corresponding expected parity distribution. Combined with the Birch–Swinnerton-Dyer conjecture and the heuristic that twists of rank at least $2$ are rare, this supports the displayed average-rank prediction; the source does not establish the conjecture itself.