Wusheng Zhu in 2012 uploaded to arxiv.org an interesting preprint titled "Riemann Zeta Function Expressed as the Di fference of Two Symmetrized Factorials Whose Zeros All Have Real Part of 1/2" (arxiv:1208.1440v2)
Let $\eta(s)$ be the Dirichlet eta function: $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^s},\quad\mathrm{Re}(s)>0$$ Then $\eta(s)+\eta(1-s)$ is conditionally convergent in the critical strip $0<\mathrm{Re}(s)<1$.
In equation (60) he expressed $\eta(s)+\eta(1-s)$ as: $$\eta(s)+\eta(1-s)=\lim_{m\to\infty} (F_m(s)-G_m(s)),\quad 0<\mathrm{Re}(s)<1 \tag{2}$$
$$F_m(s)=2\left(\sum_{k=0}^{m/2}\frac{\eta(2k+2)}{(2k)!(m-2k)!}\right)\prod_{j=1}^{m/2}\left[\left(s-\frac{1}{2}\right)^2+\Theta_j^2\right]\tag{3}$$
$$G_m(s)=2\left(\sum_{k=0}^{m/2-1}\frac{\eta(2k+3)}{(2k+1)!(m-2k-1)!}\right)\prod_{j=1}^{m/2}\left[\left(s-\frac{1}{2}\right)^2+\Phi_j^2\right]\tag{4}$$
where $\Theta_j^2,\Phi_j^2$ are positive real and $\{1/2\pm i\Theta_j\}$ are zeros of $F_m(s)$ and $\{1/2\pm i\Phi_j\}$ are zeros of $G_m(s)$.
He then mentioned that to prove
(A) that all the zeros in the critical strip $0<\mathrm{Re}(s)<1$ for $\eta(s)$ are on the critical line, it is suffice to prove
(B) that all the zeros in the critical strip $0<\mathrm{Re}(s)<1$ for $\eta(s)+\eta(1-s)$ are on the critical line. It is then suffice to prove that
(C) $$\Theta_1^2<\Phi_1^2<\Theta_2^2<\Phi_2^2<\cdots <\Theta_n^2<\Phi_n^2<\cdots \tag{5A}$$
or $$\Phi_1^2<\Theta_1^2<\Phi_2^2<\Theta_2^2<\cdots <\Phi_n^2<\Theta_n^2<\cdots \tag{5B}$$
Question 1 Assuming that (2),(3),(4) are correct, is there anything wrong or missing in this general approach?
I would guess that he needs to prove uniform convergence in the critical strip; i.e., given $0<\epsilon<1$, there exists a positive integer $M$ such that when $m>M$, $|\eta(s)+\eta(1-s)-F_m(s)+G_m(s)|<\epsilon$
Question 2 Are there similar approaches in the literature that are rigorous and also being accepted?
Update: Instead of dealing with functions of $s$, we can set $s=1/2+iz$ and deal with functions of $z$.
We define $h(z), f_m(z), g_m(z), s=1/2+ iz$ as
$$h(z^2):=\eta(s)+\eta(1-s)=\lim_{m\to\infty} (f_m(z^2)-g_m(z^2)),\quad -1/2<\mathrm{Im}(z)<1/2 \tag{2b}$$
$$f(z^2):=F_m(s)=2\left(\sum_{k=0}^{m/2}\frac{\eta(2k+2)}{(2k)!(m-2k)!}\right)(-1)^{m/2}\prod_{j=1}^{m/2}\left[z^2-\Theta_j^2\right]\tag{3b}$$
$$g_m(z^2):=G_m(s)=2\left(\sum_{k=0}^{m/2-1}\frac{\eta(2k+3)}{(2k+1)!(m-2k-1)!}\right)(-1)^{m/2}\prod_{j=1}^{m/2}\left[z^2-\Phi_j^2\right]\tag{4b}$$
where $\Theta_j^2,\Phi_j^2$ are the only and real zeros of $f_m(z)$ and $g_m(z)$.
Thus (5A) means the zeros of $f_m(z)$ strictly left-interlacing with those of $g_m(z)$ and (5B) means the zeros of $f_m(z)$ strictly right-interlacing with those of $g_m(z)$