Braun's Conjecture

Let  be an infinite Abelian semigroup with linear order  such that  is the unit element and  implies  for . Define a Möbius function  on  by  and

for , 3, .... Further suppose that  (the true Möbius function) for all . Then Braun's conjecture states that

for all .

REFERENCES:

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