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 ??