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Braun's Conjecture

Let B={b_1,b_2,...} be an infinite Abelian semigroup with linear order b_1<b_2<... such that b_1 is the unit element and a<b implies ac<bc for a,b,c in B. Define a Möbius function mu on B by mu(b_1)=1 and

sum_(b_d|b_n)mu(b_d)=0

for n=2, 3, .... Further suppose that mu(b_n)=mu(n) (the true Möbius function) for all n>=1. Then Braun's conjecture states that

b_(mn)=b_mb_n

for all m,n>=1.

See also

Möbius Problem

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References

Flath, A. and Zulauf, A. "Does the Möbius Function Determine Multiplicative Arithmetic?" Amer. Math. Monthly 102, 354-256, 1995.

Referenced on Wolfram|Alpha

Braun's Conjecture

Cite this as:

Weisstein, Eric W. "Braun's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BraunsConjecture.html

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