Möbius function μ(n): μ(1)=1, μ(n)=0 if n has a squared prime factor, μ(n)=(−1)ᵏ if n = p₁p₂...pₖ (distinct primes). Key in Möbius inversion: if g(n)=Σd|n f(d) then f(n)=Σd|n μ(d)g(n/d). Connected to ζ(s): Σμ(n)/nˢ = 1/ζ(s).
How to Use
Enter n:
- μ(n): −1, 0, or 1
- Factors: Prime factorization
- Squarefree: No repeated factors?
Möbius Inversion
If g(n) = Σ_{d|n} f(d), then f(n) = Σ_{d|n} μ(d)·g(n/d). This recovers f from its summatory function. Example: φ(n) = Σ_{d|n} μ(d)·(n/d).
Mertens Function
M(n) = Σₖ₌₁ⁿ μ(k). The Mertens conjecture |M(n)| < √n was disproved by Odlyzko & te Riele (1985). The Riemann Hypothesis is equivalent to M(n) = O(n^(1/2+ε)).
Step-by-Step Instructions
- 1Enter n.
- 2Get μ(n).
- 3View factorization.
- 4Check squarefree.
- 5See Mertens M(n).