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Mertens function : new formulation

  1. Apr 8, 2012 #1
    Is this identity true?

    Look at attachment

    Thank you.

    Attached Files:

  2. jcsd
  3. Apr 9, 2012 #2
    Example :

    M(50) = mu(27)+mu(29)+mu(31)+....+mu(49)
    = 0+(-1)+(-1)+......+0= -3

    mu(n) is Mobius function
  4. Apr 9, 2012 #3
    It is true!
    People checked it in other fora...
    So thanx for reading the post.
  5. Apr 10, 2012 #4
    The traditional definition: M(n) = mu(1)+mu(2)+...+mu(n)

    for n=8 we have M(8) = 1+(-1)+(-1)+0+(-1)+1+(-1)+0 = -2

    Your formula Gh(8) = mu(4+2)+mu(4+4) = mu(6)+mu(8) = 1

    in contradiction to the traditional formula
    Last edited: Apr 10, 2012
  6. Apr 10, 2012 #5
    n=8=2*4 an 4 is not odd

    Read the condition : o must be odd >=3 then you can compute M(2*o)

    My formula holds. Someone in another forum just proved it.
    I have a proof but it is little bit long.

    Thank you for your comment
  7. Apr 10, 2012 #6
    With n=2*o, (o odd) it's OK

    For my investigation I used ARIBAS (Windows version) and I programmed a function 'SmallMoebiusMu(n)' (small because n must not exceed 2**32) and with this function, I compared my function 'SmallMertensNumber(n)' (traditional definition) to the function 'Gaussianheart(n)' (your formulation) and for n=2,6,10,14,18,...,402 I found equal results.

    SmallMoeniusMu uses the built-in ARIBAS function 'factor16' and 'prime32test'.

    Regards from Germany
  8. Apr 18, 2012 #7
    The formula is correct!
    Good for me!
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