Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \ p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines:

Below is the plot $\sum _\rho^{100} -\frac{\cos \left(\rho \log x \ \right)}{\rho }$, where $\rho$ is the imaginary part of the non-trivial $\zeta$ zeros, and this time the $x$-values of the primes are marked with dashed lines:

In what sense is $\sum _p \frac{\cos \left(x \log p\right)}{\sqrt{p}}$ the inverse of $\sum _\rho \frac{\cos \left(\rho \log x \right)}{\rho }$ (despite the former not converging on the infinite sum)?

Also, does the first sum spike at all zeros of the Riemann $\zeta$ function, or does the scaling factor $\sqrt p$ depend on the RH?