With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is:

$$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - \sum_{\rho} \left(\dfrac{x^{\beta+ \gamma \,i}}{\beta+ \gamma \,i}+\dfrac{x^{1-\beta- \gamma \,i}}{1-\beta- \gamma \,i}\right)$$

This function counts all prime powers of the type $p^k, k \in \mathbb{N}$ residing $\le x$. I wondered what would happen to the $\rho$'s when I would divide $k$ by a real number $t \in \mathbb{R}^{+}$. Numerical evidence suggests that the following formula holds for all $t$ (i.e. counting all prime powers of the form $p^{\frac{k}{t}})$:

$$\psi(x,t) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - \sum_{\gamma} \left(\dfrac{x^{1-\frac{t}{2}+ t\,\gamma \,i}}{1-\frac{t}{2}+ t\,\gamma \,i}+\dfrac{x^{1-\frac{t}{2}- t\,\gamma \,i}}{1-\frac{t}{2}- t\,\gamma \,i}\right)$$

The graph below illustrates the point (for three values of $t$ and using the first 1000 $\rho$'s):

The adjusted real part $\beta = 1-\frac{t}{2}$ can now become any value $<1$. For instance: at $t=2$ then $\beta=0$ and $\psi(x,2)$ counts all primes, prime powers but now also its squared roots $\le x$.

Could $\psi(x,t)$ also be formally derived 'bottom-up' through Fourier analysis (and assuming the RH)?