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Does Denjoy's Probabilistic Interpretation actually "prove" that the Mertens function ratio between numbers with odd number of distinct prime factors and even number of prime factors is 1? Or does it just provide an estimate/gess based on a heuristic approach?

If not, then would one get credit if he/she actually formally prove that it does?

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  • $\begingroup$ what do you mean by the ratio between the two? if you write $M(x)=E(x)-O(x)$ then $F(x)=E(x)+O(x) =cx+o(x)$ while $M(x)=o(x)$ so $E(x)/O(x)=(F(x)+M(x))/(F(x)-M(x)) \to 1$ $\endgroup$
    – Conrad
    Commented Feb 26 at 16:18

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