# 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 ??

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This should follow from the fundamental theorem of arithmetic. The fundamental theorem of arithmetic is encoded by the von Mangoldt function:

http://www.proofwiki.org/wiki/Sum_Over_Divisors_of_von_Mangoldt_is_Logarithm

Or with the terms exponentiated as in this oeis table: http://oeis.org/A140256

Taking partial products in the vertical direction we get this oeis table: http://oeis.org/A139547

which is the same as:

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

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