$$\log^{1/2} Q \int_{-(2-\delta)}^{2-\delta} g(x) \Phi(Q^{-|x|}){\,\text{\rm d}}x \leq 2\log^{-1/2} Q \int_{Q^{-({2-\delta})}}^1 \Phi(t)\frac{{\,\text{\rm d}}t}{t} = O(\log^{-1/2} Q),$$ we can use the asymptotic estimate to obtain $$\begin{aligned}
\int_{-\infty}^{\infty}g({x})F_{\Phi}(Q^{x},W){\,\text{\rm d}}{x}
  & \leq \int_{-(2-\delta)}^{2-\delta}g({x})F_{\Phi}(Q^{x},W){\,\text{\rm d}}{x} \\ & =  g(0)+ \int_{-(2-\delta)}^{2-\delta}g({x})(1-(1-|x|)_+){\,\text{\rm d}}x + O(\log^{-1/2} Q) + o(1)
\\ & =\frac{2 {\mathcal{L}}(f)}{r(f)} + O(\delta) + o(1).\end{aligned}$$ We then conclude that $$N^*_{\Phi}(Q) \leq N_{\Phi}(Q) \left({\mathcal{L}}(f) + O(\delta) + o(1) \right).$$ Using we finish the proof.

Bounding $N_1^*(T)$
-------------------

Similarly to the case of the Riemann zeta-function, the functionals that we need to define depend on the asymptotic behavior of the function $F_1(x,T)$ defined by $$\begin{aligned}
 \label{5_9_18:57pm}
F_1(x,T)=N_1(T)^{-1}\displaystyle\sum_{0<\gamma_1, \gamma'_1 \leq T}T^{i\alpha(\gamma_1-\gamma'_1)}w(\gamma_1-\gamma'_1),\end{aligned}$$ where $x\in\mathbb{R}$, $T>0$ and the sum is over pairs of ordinates of zeros (with multiplicity) of $\xi'(s)$. To analyze $N_1^{*}(T)$ we define the following functional $$\begin{aligned}
{\mathcal{Z}}_{1}(f) =r(f)+\dfrac{2}{r(f)}\int_{0}^{r(f)}x\, f(x){\,\text{\rm d}}x-\dfrac{8}{r(f)^2}\int_{0}^{r(f)}x^2\, f(x){\,\text{\rm d}}x  +\displaystyle\sum_{k=1}^{\infty}\dfrac{2c_k}{r(f)^{2k+1}}\int_{0}^{r(f)}x^{2k+1}\, f(x){\,\text{\rm d}}x,\end{aligned}$$ where $c_k=2^{2k+1}\frac{(k-1)!}{(2k)!}$.

\[thmsimplezerosofxiprime\] Let $f\in {\mathcal{A}}_{LP}$. Assuming RH, for every fixed small $\delta >0$ we have $$N_1^*(T)\leq ({\mathcal{Z}}_1(f) + O(\delta) + o(1))N_1(T).$$

A result similar to for the function $F_1(x,T)$ defined in is also known (see [@FGL Theorem 1.1]), which is the following: for any fixed small $\delta>0$ we have $$F_1(x,T)=T^{-2|x|}\log T+|x| - 4|x|^2+\displaystyle\sum_{k=1}^{\infty}c_k |x|^{2k+1}+ o(1)(1+T^{-2|x|}\log T),$$ uniformly for $|x|\leq 1-\delta$ as $T\to\infty$, 