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].

References

[a1]	A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" Math. Comp. , 63 (1994) pp. 415–419
[a2]	D.H. Lehmer, "On Euler's totient function" Bull. Amer. Math. Soc. , 38 (1932) pp. 745–751
[a3]	V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" Nieuw Archief voor Wiskunde (4) , 5 (1987) pp. 77–83
[a4]	M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" Rocky Mount. J. Math. , 15 (1985) pp. 609–620
[a5]	R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188
[a6]	R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190
[a7]	L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155