Alphabeta Math
ExampleConstruction: AI-adaptedVerification: Not suppliedaudited 2026-07-31
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The Möbius function on the divisor poset of 12 and its agreement with μ(1),μ(2),μ(3),μ(4),μ(6),μ(12)

Example

The positive divisors of 12 are 1,2,3,4,6,12. In the divisibility order (The divisibility poset of positive integers), the covers are

1⋖2,1⋖3,2⋖4,2⋖6,3⋖6,4⋖12,6⋖12.

1[1]2[¡1]3[¡1]4[0]6[1]12[0]nodelabel:d[¹j(1;d)]

For every comparable a∣b, The number-theoretic Möbius function is the poset Möbius function of divisibility: μ(n)=μ∣(1,n) gives μ∣(a,b)=μ(b/a). Hence the full table is obtained from

μ(1)=1,μ(2)=−1,μ(3)=−1,μ(4)=0,μ(6)=1,μ(12)=0.

In particular, the row from 1 is (1,−1,−1,0,1,0) in the divisor order listed above. The recurrence checks the less immediate values: 1−1−1+μ(1,6)=0 gives μ(1,6)=1, and 1−1−1+0+1+μ(1,12)=0 gives μ(1,12)=0 (The Möbius recurrence: μP(x,x)=1 and both interval sums of μP vanish when x<y).

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