This question is related to an answer I posted earlier at the following link.
A potential explicit formula for the fundamental prime-counting function $\pi(x)$ is typically discussed in terms of Riemann's explicit formula for the prime-power counting function $\Pi(x)$ and the relationship between $\pi(x)$ and $\Pi(x)$, but I've investigated several explicit formulas analogous to (1) below where $\nu(k)$ is the number of primes in the factorization of $k$ and $M_o(x)$ is the explicit formula for the Mertens function illustrated in (2) below.
(1) $\quad\pi_o(x)=\sum\limits_{k=1}^K\nu(k)\,M_o\left(\frac{x}{k}\right)$
(2) $\quad M_o(x)=-2+\sum\limits_\rho\frac{x^{\rho}}{\rho\,\zeta'\left(\rho\right)}+\sum\limits_n\frac{x^{-2\,n}}{(-2\,n)\,\zeta'(-2 n)}$
The following figure illustrates $\pi(x)$ in blue and formula (1) for $\pi_o(x)$ in orange where formula (1) is evaluated with an upper limit of $K=x$ and formula (2) is evaluated over the first $200$ trivial zeta-zeros and non-trivial zeta-zero pairs. The red discrete portion of the plot illustrates the evaluation of formula (1) at integer values of $x$.
Figure (1): Illustration of formula (1) for $\pi_o(x)$ (orange) evaluated with an upper limit $K=x$
I've noticed the spikes exhibited in formula (1) above near integer values of $x$ can be eliminated by evaluating formula (1) with an upper limit of $K=x+1$ instead of $K=x$ as illustrated in Figure (2) below.
Figure (2): Illustration of formula (1) for $\pi_o(x)$ (orange) evaluated with an upper limit $K=x+1$
Question (1): Does formula (1) truly converge when evaluated with an upper limit of $K=x$ and/or $K=x+1$ as the number of trivial zeta zeros and non-trivial zeta-zero pairs evaluated in formula (2) increases towards $\infty$?
The upper evaluation limit $K$ in formula (1) above for $\pi_o(x)$ can't be increased to be arbitrarily larger than $x$ because the explicit formula for $M_o(x)$ defined in (2) above doesn't converge all the way down to $x=0$ which is illustrated in the following figure where formula (2) for $M_o(x)$ is evaluated over the first $200$ trivial zeta-zeros and non-trivial zeta-zero pairs (orange curve). The underlying blue reference function is $M(x)$, and the red discrete portion of the figure illustrates the evaluation of formula (2) at integer values of $x$.
Figure (3): Illustration of formula (2) for $M_o(x)$ (orange)
Question (2): What is the lower convergence bound of formula (2) for $M_o(x)$? In other words, if $M_o(x)$ converges for $x>a$, what is $a$?
Additional explicit formulas analogous to (1) above are illustrated at the following links. Some, but not all, of these explicit formulas exhibit spikes near integer values of $x$ when evaluated with an upper limit of $x$ analogous to the spikes illustrated in Figure (1) above, and I believe these spikes can often be eliminated by evaluating these explicit formulas with an upper limit of $x+1$ instead of $x$ analogous to Figure (2) above.
Explicit Formula for $\lfloor x\rfloor$
Explicit Formulas for Divisor Summatory Functions
Explicit Formula for Divisor Summatory Function Derived from Explicit Formula for $\psi(x)$
Figures (4) and (5) below illustrate formula (1) for $\pi_o(x)$ evaluated with an upper limit $K=x$ and $K=x+1$ respectively (orange curves) where the underlying explicit formula (2) for $M_o(x)$ is evaluated over the first $200$ trivial zeta-zeros and non-trivial zeta-zero pairs. Note the discontinuity at $x=6$ in Figure (4) below seems to be eliminated in Figure (5) below.
Figure (4): Illustration of formula (1) for $\pi_o(x)$ (orange) evaluated with an upper limit $K=x$
Figure (5): Illustration of formula (1) for $\pi_o(x)$ (orange) evaluated with an upper limit $K=x+1$