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PHI Function

  1. Dec 21, 2009 #1
    I'm trying to write a sound prove to proof that phi-function is multiplicative in the aspect of number theoretic function. Please comment on my proof. Thanks in advance.

    attachment.php?attachmentid=22634&stc=1&d=1261399603.jpg
     

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  2. jcsd
  3. Dec 21, 2009 #2
    Hi, one question,

    how did you step from "the sum where gcd...m and gcd...n" to "(the sum where gcd...m) times (the sum where gcd...n)" ?
     
  4. Dec 21, 2009 #3
    The passage from the first line to the second is not correct; the set of a's such that gcd(a,m) = 1 may not be disjoint from the gcd(a,n) = 1.

    One suggestion: instead of working with the sums, try using Euler's formula for [tex]\varphi[/tex]:

    [tex]\varphi\left(n\right) = n\left(\ 1 - \frac{1}{p_{1}}\right)\left(\ 1 - \frac{1}{p_{2}}\right)...\left(\ 1 - \frac{1}{p_{k}}\right)[/tex]

    Where [tex]p_{1},...,p_{n}[/tex] are the prime factors of n.
     
  5. Dec 22, 2009 #4
    attachment.php?attachmentid=22657&stc=1&d=1261465905.jpg


    Thus, the lemma is answered , so will my proof for multiplicative function to be true?
     

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  6. Dec 24, 2009 #5
    help please (=
     
  7. Dec 24, 2009 #6
    Nobody questioned the validity of gcd(a,mn) = 1 iff gcd(a,m) = gcd(a,n) = 1; this is true. It's simply the way you split the sums that's a bit unclear, and could be improved.
     
  8. Dec 24, 2009 #7
    thanks ! but that split sum is valid?
     
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