$\eta(s)+\eta(1-s)=F(s)-G(s)$ and roots of $F(s),G(s)$ are on the critical line 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 $g_m(z)$ 
 A: Re Question 2: I came up with an idea recently and uploaded A Sequence of Cauchy Sequences Which Is Conjectured to Converge to the Imaginary Parts of the Zeros of the Riemann Zeta Function which is a reformulation I believe that if this statement could be proven then the RH would be proven. It is similarly related to interlacing zeros, formulated in terms of attractive/repulsive fixed points of a dynamical system formed by a scaled function of the Hardy Z function (normalized by Omega so that the Lipschitz constant can stay bounded, see refs in linked paper). The criteria is similar to the 
Let
\begin{equation}
  Y_{n, m} (t) = \left\{ \begin{array}{ll}
    t & m = 0\\
    t + h_{n, m} \cos (\pi n) \tanh \left( \frac{Z (Y_{n, m - 1} (t))}{|
    \Omega (t) | \prod_{k = 1}^{n - 1} \tanh (Y_{n, m - 1} (t) - y_k)} \right)
    & m \geqslant 1
  \end{array} \right.
\end{equation}
denote the $m$-th iterate of the $n$-th iteration function corresponding to
the $n$-th zero of the Hardy $Z$ function where
\begin{equation}
  \Omega (t) = \left\{ \begin{array}{ll}
    1 & t = e\\
    e^{\frac{3}{4} \sqrt{\frac{\log (t)}{\log (\log (t))}}} & t \neq e
  \end{array} \right.
\end{equation}
is a lower bound for the running maximum of $| Z (s) |$
\begin{equation}
  \max_{0 \leqslant s \leqslant t} | Z (s) | > \Omega (t) \forall t \geqslant
  45.590 \ldots
\end{equation}
ensuring that
\begin{equation}
  \frac{| Z (t) |}{\Omega (t)} > 0 \forall t \geqslant 45.590 \ldots
\end{equation}
which normalizes the range of $Z (t)$ which is known to grow in both maximum
and average value as $t \rightarrow \infty$ and $h_{n, m}$ is factor which
influences the rate of convergence
\begin{equation}
  h_{n, m} = \left\{ \begin{array}{ll}
    1 & m \leqslant 2\\
    h_{n, m - 1} &  (\Delta Y^{}_{n, m - 2} (t)) = 
    (\Delta Y^{}_{n, m - 1} (t))\\
    \frac{h_{n, m - 1}}{2} & (\Delta Y^{}_{n, m - 2} (t)) \neq
    (\Delta Y^{}_{n, m - 1} (t))
  \end{array} \right.
\end{equation}
where
\begin{equation}
  \Delta Y_{n, m} (t) = Y_{n, m} (t) - Y_{n, m - 1} (t)
\end{equation}
is the $1$-st difference of the $m$-th iterate for the $n$-th zero.
Let
\begin{equation}
  c_n (\varepsilon) = \frac{Z (\max_{t \in [0, y_n]} \{ Y_{n + 1, 1} (t)
  \geqslant t \} + \epsilon) - Z (\min_{t \in [y_n, \infty]} \{ Y_{n + 1, 1}
  (t) \leqslant t \} - \epsilon)}{2 \varepsilon + \max_{t \in [0, y_n]} \{
  Y_{n + 1, 1} (t) \geqslant t \} - \min_{t \in [y_n, \infty]} \{ Y_{n + 1, 1}
  (t) \leqslant t \}}
\end{equation}
denote the Lipschitz constant then if it is always possible
to choose a small enough positive $\varepsilon$ such that $0 < c_n
(\varepsilon) < 1$ then the Riemann Hypothesis is true.
