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Number Theory : Mobius Inversion


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Define [math]\Lambda(n):=\log(p)[/math] if n is a power of a prime p

and 0 if n = 1 or n is a composite number


Prove that [math]\Lambda(n)=\sum_{d|n}\mu(\tfrac{n}{d})\log(d)[/math]


The hint says to look at [math]\sum_{d|n}\Lambda(d)[/math] and apply the Mobius inversion formula.


So far I have got [math]\sum_{d|n}\Lambda(d)= \sum_{i=1}^r \log(p_i)= \log(\prod_{i=1}^r p_i)[/math]

assuming that n has r distinct primes in its expansion.


So help :)


Don't mind the above, I have figured it out. I will post more questions if any in this thread instead.

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