Analyzing a Proof of Phi-Function Multiplicativity

[icystrike]

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.

 

[dodo]

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)" ?

 

[JSuarez]

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 $$\varphi$$:

$$\varphi\left(n\right) = n\left(\ 1 - \frac{1}{p_{1}}\right)\left(\ 1 - \frac{1}{p_{2}}\right)...\left(\ 1 - \frac{1}{p_{k}}\right)$$

Where $$p_{1},...,p_{n}$$ are the prime factors of n.

 

[icystrike]

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

 

[icystrike]

help please (=

 

[JSuarez]

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.

 

[icystrike]

thanks ! but that split sum is valid?