This question assumes the definitions of the Mellin and Fourier transforms illusrated in (1) and (2) below and the corresponding relationship illustrated in (3) below.
(1) $\quad\mathcal{M}_x[f(x)](s)=\int\limits_0^\infty f(x)\,x^{s-1}\,dx$
(2) $\quad\mathcal{F}_x[f(x)](t)=\int\limits_{-\infty}^\infty f(x)\,e^{-i\,t\,x}\,dx$
(3) $\quad\mathcal{F}_u\left[f\left(e^u\right)\right](s)=\mathcal{M}_x[f(x)](-i\,s)$
The von Mangoldt explicit formula for the second Chebyshev function and its first-order derivative are as follows.
(4) $\quad\psi_o(x)=x-\sum\limits_{k=1}^\infty\left(\frac{x^{\rho_k}}{\rho_k}+\frac{x^{\rho_{-k}}}{\rho_{-k}}\right)-\log(2\,\pi)-\sum\limits_{n=1}^\infty\frac{x^{-2\,n}}{-2\,n}\\$ $\qquad\qquad\quad=x-\sum\limits_{k=1}^\infty\left(\frac{x^{\rho_k}}{\rho_k}+\frac{x^{\rho_{-k}}}{\rho_{-k}}\right)-\log(2\,\pi)-\frac{1}{2}\log\left(1-\frac{1}{x^2}\right)$
(5) $\quad\psi_o'(x)=1-\sum\limits_{k=1}^\infty\left(x^{\rho_k-1}+x^{\rho_{-k}-1}\right)-\sum\limits_{n=1}^\infty x^{-2\,n-1}\\$ $\qquad\qquad\quad=1-\sum\limits_{k=1}^\infty\left(x^{\rho_k-1}+x^{\rho_{-k}-1}\right)+\frac{1}{x-x^3}$
I've been investigating derivation of a zeta-zero counting function based on the $f(x)$ function defined in (6) below, the Mellin transform defined in (7) below, and the subsequent integral illustrated in (8) below all of which assume the Riemann Hypothesis (as does this entire question). Note $f(x)$ defined in (6) below is related to $\psi_o'(x)$ defined in (5) above.
(6) $\quad f(x)=\sum\limits_{k=1}^\infty\left(x^{\rho_k-1}+x^{\rho_{-k}-1}\right)=\sum\limits_{k=1}^\infty\left(x^{-1/2+i\,\Im(\rho_k)}+x^{-1/2-i\,\Im(\rho_k)}\right)$
(7) $\quad F(t)=\frac{1}{2\,\pi}\mathcal{M}_x\left[x^{1/2}f(x)\right](-i\,t)=\sum\limits_{k=1}^\infty\left(\delta\left(t-\Im\left(\rho_k\right)\right)+\delta\left(t+\Im\left(\rho_k\right)\right)\right)$
(8) $\quad N(T)=\int\limits_0^T F(t)\,dt=\sum\limits_{k=1}^\infty \left(\theta\left(T-\Im\left(\rho_k\right)\right)+\theta\left(T+\Im\left(\rho_k\right)\right)\right)$
The Mellin transform illustrated in (7) above follows from the Mellin transform illustrated in (9) below which can also be understood in terms of the equivalent Fourier transform illustrated in (10) below.
(9) $\,\,\,\mathcal{M}_x\left[x^{1/2}\left(x^{-1/2+i\,\Im\left(\rho_k\right)}+x^{-1/2-i\,\Im\left(\rho_k\right)}\right)\right](-i\,t)=2\,\pi\left(\delta\left(t-\Im\left(\rho_k\right)\right)+\delta\left(t+\Im\left(\rho_k\right)\right)\right)$
(10) $\quad\mathcal{F}_u\left[\left(e^u\right)^{1/2}\left(\left(e^u\right)^{-1/2+i\,\Im\left(\rho_k\right)}+\left(e^u\right)^{-1/2-i\,\Im\left(\rho_k\right)}\right)\right](t)=\\$ $\qquad\quad\mathcal{F}_u\left[e^{u/2}\left(2\,e^{-u/2}\,\cos\left(\Im\left(\rho_k\right)\log \left(e^u\right) \right)\right)\right](t)=2\,\pi\left(\delta \left(t-\Im\left(\rho_k\right)\right)+\delta\left(t+\Im\left(\rho_k\right)\right)\right)$
The Mellin transform defined in (7) above can be approximated using finite integration limits as illustrated in (10) below, and the result can then be integrated as illustrated in (11) below to approximate the $N(T)$ function defined in (8) above.
(11) $\quad\overset{\text{~}}{F}(t)=\frac{1}{2\,\pi}\int\limits_{1/X}^X x^{1/2}\,f(x)\,x^{-i\,t-1}\,dt=\frac{i}{2\,\pi}\sum\limits_{k=1}^K\left(\frac{X^{-i\,\left(t-\Im\left(\rho_k\right)\right)}-X^{i\,\left(t-\Im\left(\rho_k\right)\right)}}{t-\Im\left(\rho_k\right)}+\frac{X^{-i\,\left(t+\Im\left(\rho_k\right)\right)}-X^{i\,\left(t+\Im\left(\rho_k\right)\right)}}{t+\Im\left(\rho_k\right)}\right)\\$ $\qquad\qquad\qquad\qquad\qquad=\sum\limits_{k=1}^K\left(\frac{\sin\left(\log(X)\,\left(t-\Im\left(\rho_k\right)\right)\right)}{\pi\,\left(t-\Im\left(\rho_k\right)\right)}+\frac{\sin\left(\log(X)\,\left(t+\Im\left(\rho _k\right)\right)\right)}{\pi\,\left(t+\Im\left(\rho _k\right)\right)}\right),\quad X\to\infty\land K\to\infty$
(12) $\quad \overset{\text{~}}{N}(T)=\int\limits_0^T\overset{\text{~}}{F}(t)\,dt=\sum\limits_{k=1}^K\left(\frac{\text{Si}\left(\log(X)\left(T-\Im\left(\rho_k\right)\right)\right)}{\pi}+\frac{\text{Si}\left(\log(X)\left(T+\Im\left(\rho_k\right)\right)\right)}{\pi}\right),\quad X\to\infty\land K\to\infty$
The following plot illustrates $\overset{\text{~}}{F}(t)$ defined in formula (11) above evaluated at $X=10,000$ and $K=10$. The red discrete portion of the plot illustrates $\frac{1}{2}\left(\overset{\text{~}}{F}\left(\Im\left(\rho_k\right)-\epsilon\right)+\overset{\text{~}}{F}\left(\Im\left(\rho_k\right)+\epsilon\right)\right)$ where $\epsilon=0.01$.
Figure (1): Illustration of $\overset{\text{~}}{F}(t)$ defined in Formula (11)
The following plot illustrates $\overset{\text{~}}{N}(T)$ defined in formula (12) above evaluated at $X=10,000$ and $K=10$. The red discrete portion of the plot illustrates $\frac{1}{2}\left(\overset{\text{~}}{N}\left(\Im\left(\rho_k\right)-\epsilon\right)+\overset{\text{~}}{N}\left(\Im\left(\rho_k\right)+\epsilon\right)\right)$ where $\epsilon=0.01$.
Figure (2): Illustration of $\overset{\text{~}}{N}(T)$ defined in Formula (12)
Question (1): Can a zeta-zero counting function be derived from a non-zeta-zero representation of the first-order derivative $\psi'(x)$ of the second Chebyshev function?
I've been attempting to derive a zeta-zero counting function based on (6), (7), and (8) above using a non-zeta-zero representation of the first-order derivative $\psi'(x)$ of the second Chebyshev function and have not yet been successful, but the approach I've been investigating is outlined below.
Note the $f(x)$ function defined in (6) above obeys the functional equation illustrated in (13) below which follows from (14) below. Note when the right-side of (14) below is simplified, the two terms on the right-side are in reverse-order of the two terms on the left-side.
(13) $\quad f(x)=\frac{1}{x}f(\frac{1}{x})$
(14) $\quad\left(x^{-1/2+i\,\Im(\rho_k)}+x^{-1/2-i\,\Im(\rho_k)}\right)=\frac{1}{x}\left(\left(\frac{1}{x}\right)^{-1/2+i\,\Im(\rho_k)}+\left(\frac{1}{x}\right)^{-1/2-i\,\Im(\rho_k)}\right)$
Assuming the definition of $g(x)$ defined in (15) below, $f(x)$ is approximated in the intervals for $x>1$ and $0<x<1$ as illustrated in (16) and (17) below. Note (16) below follows from (5) above (since $\psi'(x)$ is approximated by $\psi_o'(x)$ for $x>1$), and (17) below follows from (13) above.
(15) $\quad g(x)=1-\psi'(x)-\sum\limits_{n=1}^\infty x^{-2\,n-1}=1-\psi'(x)+\frac{1}{x-x^3}$
(16) $\quad\overset{\text{~}}{f}(x)=g(x)\,,\qquad 1<x$
(17) $\quad\overset{\text{~}}{f}(x)=\frac{1}{x}g(\frac{1}{x})\,,\quad 0<x<1$
Based on the approximations defined in (16) and (17) above, I've been thinking perhaps $F(t)$ defined in (7) above can be approximated as illustrated in (18) below (using a non-zeta-zero representation of $\psi'(x)$ in $g(x)$ defined in (15) above), and the result can then be integrated as illustrated in (19) below to approximate $N(T)$ defined in (8) above.
(18) $\quad\overset{\text{~}}{F}(t)=\frac{1}{2\,\pi}\left(\int\limits_1^\infty x^{1/2}\,g(x)\,x^{-i\,t-1}\,dx+\int\limits_0^1 x^{1/2}\left(\frac{1}{x}\,g(\frac{1}{x})\right) x^{-i\,t-1}\,dx\right)\\$ $\qquad\qquad\quad=\frac{1}{2\,\pi}\left(\int\limits_1^\infty x^{1/2}\,g(x)\,x^{-i\,t-1}\,dx+\int\limits_0^1 x^{-1/2}\,g(x^{-1})\,x^{-i\,t-1}\,dx\right)$
(19) $\quad\overset{\text{~}}{N}(T)=\int\limits_0^T\overset{\text{~}}{F}(t)\,dt$
The approach outlined above is an oversimplification since there are convergence issues that need to be addressed which are described in the two answers posted by reuns below. After reading the two answers posted by reuns below, I realized the approach I've been investigating is related to Weil's explicit formula, and the two answers posted below are probably a more correct way to approach the derivation of a zeta-zero counting function from a non-zeta-zero representation of $\psi'(x)$.
Best Answer
This is how you obtained your formula. If the series converges only in the sense of distributions you need to replace $1_{|w| < T}$ by a $C^\infty_c$ approximation of it (being careful with the $w_l$ close to $T$)
Let $\phi \in C^\infty_c([-1,1]),\int_{-\infty}^\infty \phi(w)dw = 1$ and $\psi_{T,m} = 1_{[-T,T]} \ast m\phi(m.)$ a $C^\infty_c$ approximation of $1_{[-T,T]}$ (can you replace $C^\infty_c$ by $C^1_c$ ?).
If there are no zeros on $1/2+i[T-1/m,T+1/m]$ then
$$N(T) = \sum_{|\Im(\rho)| < T} 1 = \int_{-\infty}^\infty \psi_{T,m}(w) (\sum_{t = \Im(\rho)} \delta(w- t))dw $$ $$= \frac{1}{2\pi}\int_{-\infty}^\infty \mathcal{F}^{-1}[\psi_{T,m}](u) (\sum_n \frac{\Lambda(n)}{n^{1/2}} (\delta(u-\ln n)+\delta(u+\ln n)))du - \int_{-\infty}^\infty \psi_{T,m}(w)f(w)dw$$
Approximating $e^{-u/2}\psi'(e^u)+e^{u/2}\psi'(e^{-u})$ with the approximate explicit formula for $\psi$ you get your formula again.