# Chebyshev function identity

given the Chebyshev function

$$\sum_{n \le x} \Lambda (n) = \Psi (x)$$

with $$\Lambda (n) = \log p$$ for $n=p^{k}$ and $0$ otherwise

is then true that (i think i saw it in apostol book)

$$\Psi(x) + \Psi(x/2) + \Psi (x/3)+ \ldots = \log\lfloor{x!}\rfloor$$

here $x!$ stands for factorial of '$x$'

in case the result is incorrect , what would be the correct result ??

$$\Psi(x) + \Psi(x/2) + \Psi (x/3)+ \ldots = \log\lfloor{x!}\rfloor$$