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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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up vote 4 down vote accepted

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

Or with the terms exponentiated as in this oeis table:

Taking partial products in the vertical direction we get this oeis table:

which is the same as:

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

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