Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ??

share|improve this question

1 Answer 1

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: 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:

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

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.