# Number of primitive roots modulo p; asymptotic behavior

I know that number of primitive roots modulo p is $\varphi(p-1)$, where $\varphi$ is Euler totient function. I'm actually interested in asymptotic behavior of $\frac{\varphi(p-1)}{p-1}$ (percentage of primitive roots among elements of $\mathbb{Z}_p^*$ ).

It's easy to see it attains it's maximum (0.5) on Fermat primes (those of form $2^n + 1$), and you can calculate a little bit to see what kind of primes correspond to some fractions, but that's not really interesting.

Do we know is there a limit of that sequence? Does it get arbitrarily close to zero? Is there a way we can speak of mean value of that sequence?

• There's no limit. It does get arbitrarily close to zero, because for every $k$ there are primes of the form $kn+1$ --- so choose $k$ with lots of small prime factors. – Gerry Myerson Jul 15 '13 at 9:13

A theorem is stated here: Let $D(u)$ be the relative asymptotic density in the set of all primes of the set $$\{{\,p{\rm\ prime}:\phi(p-1)/(p-1)\le u\,\}}$$ Then $D(u)$ exists for every real number $u$, $D(u)$ is a continuous function of $u$, and $D(u)$ is strictly increasing on $[0,1/2]$, with $D(0)=0$, $D(1/2)=1$. The theorem is attributed to I. Katai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), 278-289.