This question assumes the following definitions.
(1) $\quad \psi(x)=\sum\limits_{n\le x}\Lambda(n)\qquad\text{(second Chebyshev function)}$
(2) $\quad M(x)=\sum\limits_{n\le x}\mu(n)\qquad\text{(Mertens function)}$
It seems to be a fairly widely held notion that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ are related to the primes, but assuming the Riemann Hypothesis (RH) I believe the non-trivial zeta-zeros can be approximated by relationship (4) below as well as relationship (3) below. Whereas $\Lambda(n)$ in (3) below only takes on non-zero values at prime-powers, $\mu(n)$ in (4) below takes on non-zero values at all square-free integers and the magnitude of $\mu(n)$ at the primes is no different than the magnitude of $\mu(n)$ at other square-free integers. The poles of $\frac{\zeta'(s)}{\zeta(s)}$ at the non-trivial zeta-zeros are related to $\zeta(s)$ in the denominator and have nothing to do with $\zeta'(s)$ in the numerator.
(3) $\quad\frac{\zeta'(s)}{\zeta(s)}=-\underset{X\to\infty}{\text{lim}}\left(\frac{s}{s-1}+s\int_1^X(\psi(x)-x)\, x^{-s-1}\,dx\right)\\$ $\qquad\qquad=-\underset{X\to\infty}{\text{lim}}\left(\frac{s X^{1-s}}{s-1}+\sum\limits_{n=1}^X\Lambda(n)\left(n^{-s}-X^{-s}\right)\right)\,,\quad\Re(s)>\frac{1}{2}\quad\text{(assuming RH})$
(4) $\quad\frac{1}{\zeta(s)}=\int\limits_0^\infty M(x)\,x^{-s-1}\,dx=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^{N}\frac{\mu(n)}{n^s}\right),\quad \Re(s)>\frac{1}{2}\quad\text{(assuming RH})$
Furthermore, there is an explicit formula for $M(x)$ defined in (2) above as well as $\psi(x)$ defined in (1) above, but the zeta-zero terms are a bit simpler in the explicit formula for $\psi(x)$ than in the explicit formula for $M(x)$ as illustrated in (5) and (6) below.
(5) $\quad\psi_o(x)=x-\sum\limits_\rho\frac{x^\rho}{\rho}-\log(2\,\pi)+\sum\limits_n\frac{x^{-2\,n}}{2\,n}$
(6) $\quad M_o(x)=\sum\limits_\rho\frac{x^{\rho}}{\rho\,\zeta'(\rho)}-2+\sum\limits_n\frac{x^{-2\,n}}{(-2\,n)\,\zeta'(-2\,n)}$
Question: Are the non-trivial zeros of $\zeta(s)$ really related specifically to the primes, or is it more correct to say they're related to the square-free integers?