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Math Help - Mobius Inversion Formula

  1. #1
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    Mobius Inversion Formula

    PLease see the attached file.
    I understand that the formula for f(n)
    f(n) = the sum as d runs through n of (d)*F(n/d) by the inversion formula.
    The answer I came up with when solving for f(72) was 6912 so I think I am off a bit somewhere...
    mu(1)*2*72^2 + mu(2)*2*36^2 + mu(3)*2*24^2 + mu(6)*2*12^2 = f(72) = 6912
    I am not sure I am fully understanding how to use mobius inversion formula correctly so any pointers who be appreciated. Thanks for you time!
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  2. #2
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    Quote Originally Posted by mathmaniac234 View Post
    PLease see the attached file.
    I understand that the formula for f(n)
    f(n) = the sum as d runs through n of (d)*F(n/d) by the inversion formula.
    The answer I came up with when solving for f(72) was 6912 so I think I am off a bit somewhere...
    mu(1)*2*72^2 + mu(2)*2*36^2 + mu(3)*2*24^2 + mu(6)*2*12^2 = f(72) = 6912
    I am not sure I am fully understanding how to use mobius inversion formula correctly so any pointers who be appreciated. Thanks for you time!

    Putting F(n):=\sum\limits_{d\mid n}f(d)\Longrightarrow f(d)=\sum\limits_{d\mid n} \mu(d)F\left(\frac{n}{d}\right) , so directly

    f(72)=\sum\limits_{d\mid 72} \mu(72)F\left(\frac{72}{d}\right)=\mu(1)F(72)+\mu(  2)F(36)+ \mu(3)F(24)+\mu(4)F(18)+\mu(6)F(12)+\mu(8)F(9)+\mu  (9)F(8)+ \mu(12)F(6)+\mu(18)F(4)+\mu(24)F(3)+\mu(36)F(2)+\m  u(72)F(1)= F(72)-F(36)-F(24)+0+F(12)+0+0+0+0+0+0+0=

    2\cdot 72^2-2\cdot 36^2-2\cdot 24^2 + 2\cdot 12^2=6,912 and thus I get the same value you got and did the same calculations you did.

    Why do you think you're wrong?

    Tonio
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  3. #3
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    Thanks!
    I was still trying to fully understand how to utilize mobius inversion theorem properly and just naturally assumed that I was doing it incorrectly. But seeing you completely write it out has helped with my understanding so I believe we both got the right answer.

    Thanks again
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  4. #4
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    Quote Originally Posted by mathmaniac234 View Post
    Thanks!
    I was still trying to fully understand how to utilize mobius inversion theorem properly and just naturally assumed that I was doing it incorrectly. But seeing you completely write it out has helped with my understanding so I believe we both got the right answer.

    Thanks again

    Good. Of course, the basic assumption is that f(n) is an arithmetic function...

    Tonio
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