a pair correlation approach. Indeed, we will also derive functionals related to the zeros of $\xi'(s)$ and a certain average of primitive Dirichlet L-functions.

Bounding $N^*(T)$ and $N(x,T)$
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Ultimately, the functionals we need to define depend on the asymptotic behavior of $F(x,T)$. To analyze the function $N^{*}(T)$ we define the functionals $${\mathcal{Z}}(f)  =r(f) + \frac{2}{r(f)}\int_0^{r(f)}f(x)x{\,\text{\rm d}}x$$ and $${\widetilde}{\mathcal{Z}}(f)=r(f) + \frac{2}{r(f)}\int_0^{r(f)}f(x)x{\,\text{\rm d}}x + 3\int_{r(f)}^{\frac{3}{2}r(f)}f(x){\,\text{\rm d}}x -  \frac{2}{r(f)}\int_{r(f)}^{\frac{3}{2}r(f)}f(x)x{\,\text{\rm d}}x.$$

\[thmsimplezerosofxi\] Let $f\in {\mathcal{A}}_{LP}$. Assuming RH we have $$N^*(T)\leq ({\mathcal{Z}}(f)+o(1))N(T).$$ Assuming GRH, for every fixed small $\delta>0$ we have $$N^*(T)\leq ({\widetilde}{\mathcal{Z}}(f) + O(\delta) + o(1))N(T).$$

We start assuming only RH. Refining the original work of Montgomery [@M], Goldston and Montgomery [@GM Lemma 8] stated that $$\label{Fasymptotics}
F(x,T) = \big(T^{-2|x|}\log T+|x|\big)(1+o(1)),$$ uniformly for $|x|\leq 1$. Let $f\in {\mathcal{A}}_{LP}$ and let $g(x)={\widehat}f(x/r(f))/r(f)$. We can then use the explicit formula in conjunction with the asymptotic formula above to obtain $$\begin{aligned}
\sum_{0<\gamma, \gamma'\leq T}  g\bigg((\gamma-\gamma')\dfrac{\log T}{2\pi}\bigg)w(\gamma-\gamma')  = N(T)\left[\widehat{g}(0) +\int_{-1}^{1}{{\widehat}g}(x)|x|{\,\text{\rm d}}x + \int_{|x|>1}{{\widehat}g}(x)F(x,T){\,\text{\rm d}}x + o(1)
\right],\end{aligned}$$ where the $o(1)$ above is justified since ${\widehat}g$ is continuous and $T^{-2|x|}\log T\to{{\boldsymbol}\delta}_0(x)$ as $T\to\infty$ (in the distributional sense). Moreover, since $F(x,T)$ is non-negative and ${\widehat}g(x)\leq 0$ for $|x|\geq 1$ we deduce that $$\begin{aligned}
 \sum_{0<\gamma, \gamma'\leq T}g\bigg((\gamma-\gamma')\dfrac{\log T}{2\pi}\bigg)w(\gamma-\gamma') &  \leq 