Gaussian-prime zero-statistics form-factor conjecture

Let $f$ and $\Phi$ be the smooth test functions above, let $\Xi_k$ denote the relevant Hecke $L$-function, and write

$$\mathcal S_n(\Xi_k)=\sum_j\widetilde\Phi\left(\frac12+i\gamma_{k,j}\right)e^{2\pi i n\gamma_{k,j}},$$

where the sum runs over the nontrivial zeros $\frac12+i\gamma_{k,j}$ of $L(s,\Xi_k)$. For fixed $\alpha>0$, set $n=\frac{\alpha}{2}\frac{\log K}{\pi}$.

Gaussian-prime zero-statistics form-factor conjecture. As $K\to\infty$,

$$\frac{2}{K}\sum_{k>0}\widehat f\left(\frac{k}{K}\right)^2\left|\mathcal S_n(\Xi_k)\right|^2\sim c_2(f,\Phi)\log K\min(\alpha,2).$$

This averaged zero-statistics statement is presented as sufficient, at a heuristic level, to imply the smooth Gaussian-prime variance conjecture. It concerns the mean square of sums over zeros across the family of Hecke $L$-functions and remains open in the source.