Totient function

Euler totient function, Euler totient

Another frequently used named for the Euler function , which counts the natural numbers that are relatively prime to .

The Carmichael conjecture on the Euler totient function states that if , then for some ; i.e. no value of the Euler function is assumed once. This has now been verified for , [a1].

A natural generalization of the Euler totient function is the Jordan totient function , which counts the number of -tuples , , such that . Clearly, .

One has

where runs over the prime numbers dividing , and

where is the Möbius function and runs over all divisors of . For these formulas reduce to the well-known formulas for the Euler function.

The Lehmer problem on the Euler totient function asks for the solutions of , , [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 ) is easy, [a4]: For , if and only if is a prime number. Moreover, if is a prime number, then .

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 " 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 for which " 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