# Is Euler's totient function a primitive recursive function?

We consider the function $g$ which associates the number of prime integers with $n$ in the set $\{0,...,n\}$. I have to prove that $g$ is a primitive recursive function.

First I defined the set $A=\{i∈\{0,...,n\}:\gcd(i,n)=1\}$. To show that $A$ is a primitive recursive set I said that : $\{0,...,n\}$ is primitive recursive because ${k}$ is primitive recursive by characteristic function and $\{0,...,n\}=\bigcup\limits_{k=0}^{n} \{k\}$ is primitive recursive by the property of $⋃$.

Then the function $\gcd$ is primitive recursive. Indeed we can see that with the function $lcm$. We have $lcm(a,b)=μk≤ab (∃j,l≤ab,aj=k∧bl=k)$ which is a primitive recursive expression. Then by operator division $\gcd$ is a primitive recursive function.

So $A$ is a primitive recursive set.

Now, to prove that $g$ is primitive recursive can I take :

$g(i)=\sum\limits_{i=0}^{n} \chi_{A}(i)$ with $χ_A(i)=1$ if $i∈A$.and $g$ is a finite sum of primitive recursive functions so it is primitive recursive?

• We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. Jun 17, 2016 at 20:12
• @Raphael Ok I will try to modify Jun 17, 2016 at 22:20

A computable function is primitive recursive if and only if it can be computed by a program whose running time is upper-bounded by some primitive recursive function. Your function (usually known as Euler's totient function $$\varphi$$) can be computed in exponential time, and so is primitive recursive.