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 |
Totient function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totient_function&oldid=35641