Proving Mobius Behaving Pair with Möbius Function μ(n)

  • Thread starter Gear300
  • Start date
  • Tags
    Pair
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.
  • #1
Gear300
1,213
9
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.
 
Mathematics news on Phys.org
  • #2
Can you show your proof?
 

FAQ: 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.

Similar threads

I Are there any two pairs of integers with the same result in a specific function?
Replies
7
Views
1K
B Before understanding theorems in elementary Euclidean plane geometry
Replies
12
Views
2K
I Alternating Harmonic Numbers are cool, spread the word!
Replies
4
Views
1K
Ranking the pairs in a bridge match - what precision?
Replies
18
Views
3K
Prove that ## a^{({2}^{n})}\equiv 1\pmod {2^{n+2}} ## for any ##n##?
Replies
6
Views
2K
MHB A PHP Function To Perform Nth-Order Lagrange Interpolation
Replies
1
Views
4K
MHB Logic Problems: Improve Deductive Skills with Basic Totient Function
Replies
2
Views
2K
MHB Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order
Replies
2
Views
1K
Prove that ##5 | (2^n a) \rightarrow (5 | a)##for any ##n##
Replies
10
Views
858
I Proof involving functional graphs and the injective property
Replies
3
Views
966

Hot Threads

Recent Insights

Back
Top