Proving Mobius Behaving Pair with Möbius Function μ(n)

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SUMMARY

The discussion centers on proving the relationship between two functions, f and g, in the context of the Möbius function μ(n). The participants confirm that if certain conditions are met, then f and g behave similarly to a Möbius pair. The proof provided by the original poster is deemed sound, and there is an inquiry about the validity of the converse statement regarding the behavior of f and g. The conversation emphasizes the importance of rigorous proof in number theory.

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Hello friends from afar.

Given the Möbius function μ(n), prove that if

upload_2016-12-19_19-31-36.png
,

then

upload_2016-12-19_19-33-15.png
.

(The upper bound for both sums is the integer floor of x.)

I've done the proof and it seems sound, but it also seems that the converse statement is true, implying that f and g should behave similar to a Mobius pair. But there was no question for the converse statement, so I just wanted confirmation. Many thanks.
 
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