Let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental theorem of arithmetic, we have the identity: $$\log(\left \lfloor x \right \rfloor!)=\sum_{n=1}^{\infty}\psi\left(\frac{x}{n}\right)$$
Is there a way to utilize the well-known explicit formula
$$\psi(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\log(2\pi)-\frac{1}{2}\log\left(1-x^{-2}\right)\qquad\zeta(\rho)=0\;(0<\Re(\rho)<1)$$ to express $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros $\rho$?
Please notice that $\psi\left(\frac{x}{n}\right)=0$ for $n>\frac{x}{2}$. Thus, the summation might be truncated at $\left \lfloor \frac{x}{2} \right \rfloor$.
Now posted on Mathoverflow: https://mathoverflow.net/questions/149932/expressing-log-left-lfloor-x-right-rfloor-in-terms-of-zeta-zeros?rq=1