N(T).$$ Assuming GRH we have $$N({\widetilde}{\mathcal{P}}(f)+\varepsilon,T)\gg N(T).$$

In the following we only exhibit the proof assuming RH since under GRH the proof is very similar, and the only extra information needed is in . Let $f\in{\mathcal{A}}_{LP}$ and $\lambda>0$. Applying the explicit formula for $g(x)= f(r(f)x/{\lambda})$ in conjunction with we obtain $$\begin{aligned}
\sum_{0<\gamma, \gamma'\leq T}g\bigg((\gamma-\gamma')\dfrac{\log T}{2\pi}\bigg)w(\gamma-\gamma') = N(T)\int_{-\infty}^{\infty}\widehat{g}(x)F(x,T){\,\text{\rm d}}x
& \geq N(T)\left[{\widehat}g(0)+2\int_{0}^{1}{\widehat}g(x)x{\,\text{\rm d}}x+o(1)\right]
\\ & = N(T)\left[1+p_f({\lambda})+o(1)\right].\end{aligned}$$ Since ${\widehat}f\geq 0$, we have $\|f\|_{\infty}=f(0)=1$. Recall now the pair correlation function $N(x,T)$ defined in . We have $$\begin{aligned}
\sum_{0<\gamma, \gamma'\leq T}g\bigg((\gamma-\gamma')\dfrac{\log T}{2\pi}\bigg)w(\gamma-\gamma') & = N^*(T) +2\sum_{\substack{0<\gamma, \gamma'\leq T\\ 0<\gamma-\gamma'}}f\bigg((\gamma-\gamma')\dfrac{r(f)\log T}{2\pi{\lambda}}\bigg)w(\gamma-\gamma')  \\
& \leq N^*(T) +2\sum_{\substack{0<\gamma, \gamma'\leq T\\ 0<\gamma-\gamma' \leq \frac{2\pi {\lambda}}{\log T}}}f\bigg((\gamma-\gamma')\dfrac{r(f)\log T}{2\pi{\lambda}}\bigg)w(\gamma-\gamma') \\
& \leq N^*(T)+2N({\lambda},T) \\ & = (1+o(1))N(T) + 2N({\lambda},T),\end{aligned}$$ where in the last step we have used . Then, we obtain $$\begin{aligned}
 \frac{N({\lambda},T)}{N(T)}\geq \frac{p_f({\lambda})}{2}+o(1).\end{aligned}$$ Noting that $N({\lambda},T)$ increases with ${\lambda}$, we can then choose ${\lambda}$ arbitrarily close to ${\mathcal{P}}(f)$ and obtain the desired result.

Bounding $N_{\Phi,s}(Q)$
------------------------

Define the following functional over ${\mathcal{A}}_{LP}$: $${\mathcal{L}}(f) = \frac{r(f)}{2} + \dfrac{4}{r(f)}\int_{0}^{\frac{r(f)}{2}}x f(x){\,\text{\rm d}}x+2\int_{\frac{r(f)}{2}}^{r(f)}f( x){\,\text{\rm d}}x.$$ We have the following theorem.

\[thmnphi\] Let $f\in{\mathcal{A}}_{LP}$. Assuming GRH, for every fixed small $\delta>0$ we have $$N_{\Phi,s}(Q) \geq (2- {\mathcal{L}}(f) + O(\delta) + o(1))N_{\Phi}(Q).$$

For $Q>1$ 