Proving Mobius Behaving Pair with Möbius Function μ(n)

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In summary, the Mobius Behaving Pair is a pair of positive integers that satisfies the property of the Mobius function μ(n) being equal to μ(ab) for all positive integers n. This pair is significant in proving certain number theory conjectures, and is closely related to the Möbius function μ(n). One can prove the existence of a Mobius Behaving Pair using the Chinese Remainder Theorem and the multiplicative property of μ(n). There are known examples of a Mobius Behaving Pair, such as the pair (2, 3) where μ(2) = μ(3) = -1 and μ(6) = μ(2x3) = 1, thus satisfying the necessary condition.
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Gear300
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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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Can you show your proof?
 

Related to Proving Mobius Behaving Pair with Möbius Function μ(n)

1. What is the Mobius Behaving Pair?

The Mobius Behaving Pair refers to a pair of positive integers (a, b) such that the Mobius function μ(n) is equal to μ(ab) for all positive integers n.

2. What is the significance of the Mobius Behaving Pair?

The Mobius Behaving Pair is significant because it helps in proving certain number theory conjectures, such as the Goldbach's Conjecture and the Riemann Hypothesis.

3. How is the Mobius Behaving Pair related to the Möbius function μ(n)?

The Mobius Behaving Pair is closely related to the Möbius function μ(n) as it is defined based on the behavior of μ(n). Specifically, it is a pair of positive integers that satisfies the property of μ(ab) = μ(n) for all positive integers n.

4. How can one prove the existence of a Mobius Behaving Pair?

To prove the existence of a Mobius Behaving Pair, one can use the Chinese Remainder Theorem and the fact that the Möbius function μ(n) is multiplicative. By finding a suitable pair of positive integers that satisfies the necessary conditions, one can prove the existence of a Mobius Behaving Pair.

5. Are there any known examples of a Mobius Behaving Pair?

Yes, there are known examples of a Mobius Behaving Pair. One example is the pair (2, 3), where μ(2) = μ(3) = -1 and μ(6) = μ(2x3) = μ(2)μ(3) = (-1)(-1) = 1, thus satisfying the necessary condition.

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