# Distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$

Maybe this is a well-know problem.

What do we know about distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$? (Where $\phi$ is the Euler's totient function).

In the other words does there exist a distribution like $\mu$ concentrated on $[0,1]$ that for every interval $[a,b]$ we have the following equation?

$$\lim_{n\to \infty} \frac{\# \ \{i \ |\ \frac{\phi(i)}{i} \in [a,b], \ 1\leq i \leq n \}}{n} = \mu([a,b])$$

Is the sequence equidistributed in $[0,1]$?

• The existence of a distribution is well known -- see old works of Schoenberg, and Erdos and Wintner. It is certainly not equidistribution: for example note that all even numbers, and all odd multiples of $105$ will have $\phi(n)/n <1/2$. – Lucia Apr 29 '15 at 20:32
• Starting from $\frac{\phi(n)}{n} = \exp( \sum_p \log(1-\frac{1}{p}) 1_{p|n} )$, one can show (using the absolute convergence of $\sum_p \log(1-\frac{1}{p}) / p$) that the limiting distribution $\mu$ of $\frac{\phi(n)}{n}$ is the distribution of $\exp( \sum_p \log(1-\frac{1}{p}) I_p )$, where the $I_p$ are independent Bernoulli variables with expectation $1/p$. In other words, $\mu$ is the exponential of a certain Bernoulli convolution, which one would then typically expect to be rather singular in nature (as indicated by the references given in other answers here). – Terry Tao Apr 29 '15 at 22:24
• If one were interested in the limiting distribution of $\frac{\phi(p-1)}{p-1}$ instead of $\frac{\phi(n)}{n}$, the only difference would be that the $I_p$ now have expectation $1/(p-1)$ rather than $1/p$. – Terry Tao Apr 29 '15 at 22:27
• But would this approach lead to an explicit formula for the distribution or merely show its existence? – Captain Darling Apr 29 '15 at 23:05
• Bernoulli convolutions are reasonably explicit (their characteristic function is a Riesz product, so their distribution is the inverse Fourier transform of a Riesz product), although the formula might not be terribly tractable in practice. – Terry Tao Apr 30 '15 at 5:32

As Lucia points out the existence of a limiting distribution for $\phi(n)/n$ is well-known. The most direct way to show that there is a limiting distribution is to compute the moment $$\sum_{n \leq x} \Big ( \frac{\phi(n)}{n} \Big )^{k}$$ for all fixed positive integers $k$. The limiting distribution $G(t)$ is continuous but not differentiable. It is also known to be purely singular, so that $G'(t) = 0$ almost everywhere in $[0,1]$. In addition, for the modulus of continuity we know that $$\sup_{0 \leq t \leq 1} |G(t) - G(t - \varepsilon t)| \ll \frac{1}{\log(1/\varepsilon)}$$ and that this is optimal. Finally the behavior as $t \rightarrow 0$ and $t \rightarrow 1$ is markedly different. As $t \rightarrow 0$ there is a doubly exponential decay in $1/t$, while the behavior at $t \rightarrow 1$ is such that $1 - G(1 - 1/\sigma) \sim c/\log \sigma$ as $\sigma \rightarrow \infty$.
• The post of guestCLT is rather informative. I was wondering whether there is an explicit distribution for the sequence $\frac{\phi(p-1)}{p-1}$ as $p$ ranges over primes ? – Captain Darling Apr 29 '15 at 21:14