# Analyzing a Proof of Phi-Function Multiplicativity

• icystrike
In summary, the conversation revolves around proving that the phi-function is multiplicative in terms of number theoretic functions. The speaker asks for comments on their proof and receives a suggestion to use Euler's formula for phi. There is discussion about the validity of splitting the sums and the speaker receives confirmation that their proof is correct, but the splitting of the sums could be improved.
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.

#### Attachments

• mul.jpg
17.1 KB · Views: 885
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)" ?

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.

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

#### Attachments

• lemma.JPG
63.8 KB · Views: 541

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.

thanks ! but that split sum is valid?

## 1. What is the "Phi-Function" in mathematics?

The Phi-Function, also known as the Euler's totient function, is a mathematical function that counts the number of positive integers less than or equal to a given integer that are relatively prime to it. It is denoted by the symbol φ (n).

## 2. What does "multiplicativity" mean in the context of Phi-Function?

Multiplicativity refers to the property of a function that states the value of the function for a product of two numbers is equal to the product of the function values for each number individually. In the context of Phi-Function, multiplicativity means that the value of φ (ab) is equal to the product of φ (a) and φ (b) for any two relatively prime numbers a and b.

## 3. How is a proof of Phi-Function multiplicativity analyzed?

The proof of Phi-Function multiplicativity is analyzed by examining the steps and logic used to show that the function follows the property of multiplicativity. This includes understanding the definitions and properties of the Phi-Function, as well as the mathematical principles and techniques used in the proof.

## 4. What is the significance of proving Phi-Function multiplicativity?

Proving Phi-Function multiplicativity is significant because it shows that the function follows a fundamental property in mathematics and can be used to simplify calculations and solve various number theory problems. It is also a key step in proving more complex theorems and results related to the Phi-Function.

## 5. Are there any real-world applications of Phi-Function multiplicativity?

Yes, there are many real-world applications of Phi-Function multiplicativity, particularly in the field of number theory. It is used in cryptography to generate secure keys and in coding theory to construct error-correcting codes. It also has applications in primality testing, factoring large integers, and analyzing the distribution of prime numbers.

• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
15
Views
4K
• Linear and Abstract Algebra
Replies
17
Views
1K
• Calculus
Replies
3
Views
491
• Linear and Abstract Algebra
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
8
Views
870
• Calculus and Beyond Homework Help
Replies
5
Views
489
• Calculus and Beyond Homework Help
Replies
15
Views
586
• Linear and Abstract Algebra
Replies
3
Views
2K
• Topology and Analysis
Replies
8
Views
412