Weak pair correlation conjecture for Hilbert characters

Let $D<0$ be a fundamental discriminant, let $H(D)$ be the ideal class group of $\mathbb{Q}(\sqrt{D})$, and let $\hat{H}(D)$ be the family of unramified Hecke characters associated with $H(D)$. Let $w$ be a fixed smooth weight function supported in $[1,2]$, with Mellin transform

$$\hat{w}(s):=\int_{0}^{\infty}x^s w(x)\frac{dx}{x}.$$

Assume GRH holds for $L(s,\chi)$ for every $\chi\in\hat{H}(D)$, and let $T>D^{\alpha}$ for some $\alpha>0$. Weak pair correlation conjecture. One has

$$\frac{1}{h(D)}\sum_{\chi\in \hat{H}(D)}\sum_{\gamma_{\chi},\gamma_{\chi}^{\prime}}T^{i(\gamma_{\chi}-\gamma_{\chi}^{\prime})}\hat{w}(1/2+i\gamma_{\chi})\overline{\hat{w}(1/2+i\gamma_{\chi}^{\prime})}\ll \log(D),$$

where $1/2+i\gamma_{\chi}$ and $1/2+i\gamma_{\chi}^{\prime}$ are zeros of $L(s,\chi)$. The implied constant is independent of $\alpha$ and $D$ and depends only on $w$. This is a weak form of the pair correlation conjecture for the family of Hilbert characters associated with $\mathbb{Q}(\sqrt{D})$; the paper uses it to improve the bound for the number of ideal classes not represented by small primes, but does not establish it.