Möbius Function Applications

Number Theory · Axiom Academy

EXAMPLE Möbius Function Applications Computing μ(n), applying Möbius inversion, and deriving Euler's totient function Excellent work! You've completed this Möbius function example. Here's what we learned: Computing μ(n): The Möbius function equals 0 if n has a squared prime factor, 1 if n has an even number of distinct prime factors, and -1 if n has an odd number of distinct prime factors. Möbius Inversion: If f(n) = Σ d|n g(d), then g(n) = Σ d|n μ(d)f(n/d). This powerful technique allows us to invert divisor sums. Deriving φ(n): Using the identity n = Σ d|n φ(d) and Möbius inversion, we can express φ(n) = Σ d|n μ(d)(n/d), which leads to the product formula φ(n) = n∏ p|n (1 - 1/p). The Möbius function is fundamental in number theory, appearing in formulas for counting primitive roots, computing the Riemann zeta function, and solving Diophantine equations. Master these techniques to unlock deeper results!

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