Möbius-Funktion Rechner
Berechne μ(n), prüfe ob quadratfrei und entdecke die Mertens-Funktion
Die Mertens-Funktion M(n) = Σ μ(k) für k=1 bis n. Es wird vermutet (aber unbewiesen), dass |M(n)| langsamer wächst als √n.
Die ersten 50 quadratfreien Zahlen:
Definition der Möbius-Funktion
| n | Faktorisierung | μ(n) | Grund |
|---|---|---|---|
| 1 | 1 | +1 | Nach Definition |
| 2 | 2 | −1 | 1 verschiedene Primzahl |
| 4 | 2² | 0 | Hat quadratischen Faktor 2² |
| 6 | 2 × 3 | +1 | 2 verschiedene Primzahlen |
| 30 | 2 × 3 × 5 | −1 | 3 verschiedene Primzahlen |
| 210 | 2×3×5×7 | +1 | 4 verschiedene Primzahlen |
Häufig gestellte Fragen
Wofür wird die Möbius-Funktion verwendet?
The Möbius function appears in the Möbius inversion formula, which lets you invert summatory functions in number theory. It also plays a key role in the inclusion-exclusion principle, the Riemann zeta function (via the Dirichlet series 1/ζ(s) = Σ μ(n)/n^s), and the distribution of primes.
Was bedeutet "quadratfrei"?
A positive integer is squarefree if no perfect square other than 1 divides it. Equivalently, it has no repeated prime factors. For example: 6 = 2×3 is squarefree, but 12 = 2²×3 is not. Squarefree numbers are exactly those where μ(n) ≠ 0.
Was ist die Mertens-Funktion?
The Mertens function M(n) = Σ μ(k) for k from 1 to n. It oscillates around zero and grows very slowly. The famous Mertens conjecture (disproved in 1985) claimed |M(n)| ≤ √n for all n — it's actually false, but the bound holds for all tested values up to about 10^22.
Ist die Möbius-Funktion multiplikativ?
Yes! The Möbius function is completely multiplicative: if gcd(m, n) = 1, then μ(mn) = μ(m)·μ(n). This property makes it a powerful tool in multiplicative number theory and allows efficient computation using sieves.
Welcher Anteil der ganzen Zahlen ist quadratfrei?
Asymptotically, the density of squarefree integers is 6/π² ≈ 60.79%. This is derived from the probability that a random integer is not divisible by any perfect square p², using the Euler product formula.
Was sagt uns μ(n) = 0?
μ(n) = 0 means n has at least one prime factor appearing twice or more (i.e., n is not squarefree). These numbers are "filtered out" by the Möbius function, which is why it works so well for inclusion-exclusion in number-theoretic sieves.
Was ist die Möbiussche Umkehrformel?
If f(n) = Σ g(d) for all divisors d of n, then g(n) = Σ μ(n/d)·f(d). This is the number-theoretic analog of the fundamental theorem of calculus, allowing "inversion" of summatory functions over divisors.
Wer war August Möbius?
August Ferdinand Möbius (1790–1868) was a German mathematician and astronomer. He is famous for the Möbius strip (a one-sided surface), the Möbius function in number theory, and contributions to projective geometry. The function bearing his name was actually introduced in his 1832 paper on number theory.