# Totient function

Euler totient function, Euler totient

Another frequently used named for the Euler function $\phi(n)$, which counts a reduced system of residues modulo $n$: the natural numbers $k \in \{1,\ldots,n\}$ that are relatively prime to $n$.

The Carmichael conjecture on the Euler totient function states that if $\phi(x) = m$ for some $m$, then $\phi(y) = m$ for some $y \neq x$; i.e. no value of the Euler function is assumed once. This has now been verified for $x < 10^{1000000}$, [a1].

A natural generalization of the Euler totient function is the Jordan totient function $J_k(n)$, which counts the number of $k$-tuples $(a_1,\ldots,a_k)$, $a_i \in \{1,\ldots,n\}$, such that $\mathrm{hcf}\{n,a_1,\ldots,a_k\} = 1$. Clearly, $J_1 = \phi$. The $J_k$ are multiplicative arithmetic functions.

One has $$J_k(n) = n^k \prod_{p|n} \left({ 1 - p^{-k} }\right)$$ where $p$ runs over the prime numbers dividing $n$, and $$J_k(n) = \sum_{d | n} \mu(n/d) d^k$$ where $\mu$ is the Möbius function and $d$ runs over all divisors of $n$. For $k=1$ these formulae reduce to the well-known formulae for the Euler function.

The Lehmer problem on the Euler totient function asks for the solutions of $M.\phi(n) = n-1$, $M \in \mathbb{N}$, [a2]. For some results on this still (1996) largely open problem, see [a3] and the references therein. The corresponding problem for the Jordan totient function (and $k>1$) is easy, [a4]: For $k>1$, $J_k(n) | n^k-1$ if and only if $n$ is a prime number. Moreover, if $n$ is a prime number, then $J_k(n) = n^k-1$.

For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [a5], [a6].