Exercise · ZW-MOBIUS-INVERSION-FORMULA-003 · ★☆☆ Basic
Over which set is the sum taken in \(g(n)=\sum_{d\mid n}f(d)\)?
Choose one option.
Show answer
- A. The positive divisors of \(n\)
Explanation
The notation \(d\mid n\) means that \(d\) runs over the positive divisors of \(n\), including \(1\) and \(n\). None of the other sets matches that.
Review the concept: Möbius inversion formula
Formula that recovers an arithmetic function \(f\) from its sum over divisors, using the Möbius function \(\mu\).
\[f(n)=\sum_{d\mid n}\mu(d)\,g\!\left(\frac{n}{d}\right)\]
More practice problems: möbius inversion formula
- ZW-MOBIUS-INVERSION-FORMULA-001 Which arithmetic function appears as the coefficient in the Möbius inversion formula?
- ZW-MOBIUS-INVERSION-FORMULA-002 What is the correct way to invert g(n)=∑_d| nf(d)?
- ZW-MOBIUS-INVERSION-FORMULA-004 If g(n)=∑_d| nμ(d) for every n, what is g(12)?
- ZW-MOBIUS-INVERSION-FORMULA-005 What is μ(4)?
- ZW-MOBIUS-INVERSION-FORMULA-006 If g(n)=n for all n, compute f(6) using Möbius inversion.
- ZW-MOBIUS-INVERSION-FORMULA-007 If g(n)=n^2 for all n, compute f(3) using Möbius inversion.
- ZW-MOBIUS-INVERSION-FORMULA-008 If g(n)=∑_d| nf(d) and f(n)=n for all n, compute g(4).
- ZW-MOBIUS-INVERSION-FORMULA-009 Let g(n)=n for all n. What is f(12) by Möbius inversion?
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