You know that g(n)=∑_d| n f(d) for every n, with g(1)=2, g(2)=5, g(3)=4 and g(6)=9. What is f(6)?

Exercise · ZW-MOBIUS-FUNCTION-006 · ★★★ Advanced

You know that \(g(n)=\sum_{d\mid n} f(d)\) for every \(n\), with \(g(1)=2\), \(g(2)=5\), \(g(3)=4\) and \(g(6)=9\). What is \(f(6)\)?

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A. 2

Explanation

By Möbius inversion, \(f(n)=\sum_{d\mid n}\mu(d)\,g(n/d)\); for \(n=6\) this gives \(g(6)-g(3)-g(2)+g(1)=9-4-5+2=2\).

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Review the concept: Möbius function

The Möbius function \(\mu(n)\) is an arithmetic function: it equals \(1\) if \(n=1\), \((-1)^k\) if \(n\) is a product of \(k\) distinct primes, and \(0\) if \(n\) has a repeated prime factor.

\[\sum_{d\mid n}\mu(d)=\begin{cases}1 & n=1\\ 0 & n>1\end{cases}\]

You know that \(g(n)=\sum_{d\mid n} f(d)\) for every \(n\), with \(g(1)=2\), \(g(2)=5\), \(g(3)=4\) and \(g(6)=9\). What is \(f(6)\)?

Explanation

Review the concept: Möbius function

More practice problems: möbius function