Totient Function

The totient function , also called Euler's totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) , where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function can be simply defined as the number of totatives of . For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so . The totient function is implemented in Mathematica as EulerPhi[n].

is always even for . By convention, , although Mathematica defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of for , 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (Sloane's A000010). The totient function is given by the Möbius transform of 1, 2, 3, 4, ... (Sloane and Plouffe 1995, p. 22). is plotted above for small .

For a prime ,

(1)

since all numbers less than are relatively prime to . If is a power of a prime, then the numbers that have a common factor with are the multiples of : , , ..., . There are of these multiples, so the number of factors relatively prime to is

			(2)
			(3)
			(4)

Now take a general divisible by . Let be the number of positive integers not divisible by . As before, , , ..., have common factors, so

			(5)
			(6)

Now let be some other prime dividing . The integers divisible by are , , ..., . But these duplicate , , ..., . So the number of terms that must be subtracted from to obtain is

			(7)
			(8)

and

			(9)
			(10)
			(11)

By induction, the general case is then

			(12)
			(13)

where the product runs over all primes dividing . An interesting identity relating to is given by

(14)

(A. Olofsson, pers. comm., Dec. 30, 2004).

Another identity relates the divisors of to via

(15)

The totient function is connected to the Möbius function through the sum

(16)

where the sum is over the divisors of , which can be proven by induction on and the fact that and are multiplicative (Berlekamp 1968, pp. 91-93; van Lint and Nienhuys 1991, p. 123).

The totient function has the Dirichlet series generating function

(17)

for (Hardy and Wright 1979, p. 250).

The totient function satisfies the inequality

(18)

for all except and (Kendall and Osborn 1965; Mitrinović and Sándor 1995, p. 9). Therefore, the only values of for which are , 4, and 6. In addition, for composite ,

(19)

(Sierpiński and Schinzel 1988; Mitrinović and Sándor 1995, p. 9).

also satisfies

(20)

where is the Euler-Mascheroni constant. The values of for which are given by 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, ... (Sloane's A100966).

The divisor function satisfies the congruence

			(21)
			(22)

for all primes and no composite with the exception of 4, 6, and 22, where is the divisor function. This fact was proved by Subbarao (1974), despite the implication to the contrary, "is it true for infinitely many composite ?," stated in Guy (1994, p. 92), a query subsequently removed from Guy (2004, p. 142). No composite solution is currently known to

(23)

(Honsberger 1976, p. 35).

A corollary of the Zsigmondy theorem leads to the following congruence,

(24)

(Zsigmondy 1882, Moree 2004, Ruiz 2004ab).

The first few for which

(25)

are given by 1, 3, 15, 104, 164, 194, 255, 495, 584, 975, ... (Sloane's A001274), which have common values , 2, 8, 48, 80, 96, 128, 240, 288, 480, ... (Sloane's A003275).

The only for which

(26)

is , giving

(27)

(Guy 2004, p. 139).

Values of shared among that are close together include

			(28)
			(29)
			(30)
			(31)

(Guy 2004, p. 139). McCranie found an arithmetic progression of six numbers with equal totient functions,

(32)

as well as other progressions of six numbers starting at 1166400, 1749600, ... (Sloane's A050518).

If the Goldbach conjecture is true, then for every positive integer , there are primes and such that

(33)

(Guy 2004, p. 160). Erdős asked if this holds for and not necessarily prime, but this relaxed form remains unproven (Guy 2004, p. 160).

Guy (2004, p. 150) discussed solutions to

(34)

where is the divisor function. F. Helenius has found 365 such solutions, the first of which are 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, ... (Sloane's A001229).

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