2014年9月3日 · If is the Euler's Totient Function, then the proof goes as follows : By definition if p is prime, Now, if is any composite number, then , where and are prime. To see that , consider that the set of positive integers less than is the set . The integers in this set that are not relatively prime to are the set and the set . Accordingly Then as ...

Seeking for a proof on the relation φ(n) = ∑ d | n dμ(n d), between Euler totient and Möbius function

I am working on a question for my number theory class that asks: Prove that for every integer , ϕ(n) ≥ √n √2. However, I was searching around Google, and on various websites I have found people explaining that the phi function has a defined upper bound, but no lower bound. Am I reading these sites incorrectly, or am I missing something in the problem itself?

The asker asks for a proof why the totient function gives an even number. I guess I wrote a proof not a clarification.

Edited:Sorry that my question is to prove the two equation before, because I've already known how to relate the problem with these two equations, and I've also known how to prove the second equation from Identity involving Euler's totient function: ∑ k=1n ⌊n k⌋ φ(k) = n(n+1) 2 ∑ k = 1 n ⌊ n k ⌋ φ (k) = n (n + 1) 2 but I can't solve the first one with the similar method.

2017年6月1日 · When I search for Inverse of Euler's totient function I get answers for how to solve ϕ(n) = k ϕ (n) = k, which is not what I'm looking for, so maybe I'm asking the wrong question?

Not the answer you're looking for? Browse other questions tagged elementary-number-theory totient-function multiplicative-function .

More precisely, for any number field k, k, if we define ϕ(a) ϕ (a) as the number of residue classes a¯ ∈ok/a a ¯ ∈ o k / a such that gcd (a¯,a) =ok (a ¯, a) = o k. Then is the fuction ϕ(a) ϕ (a) sharing the basic properties that the usual Euler 's totient function has? For example, for all α ∈ ok α ∈ o k prime to a a we have

I know Euclid's clever way of proving that there are infinitely many primes but while I was studying, I also saw a sentence on the book: "Euler-phi function can also be used to show that there are

Let Q be the set of square-free integers less than some n and let ϕ denote the Euler totient. I am currently reading an article where it is mentioned that :