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 the Wolfram Language as EulerPhi[n].

The number  is called the cototient of  and gives the number of positive integers  that have at least one prime factor in common with .

 is always even for . By convention, , although the Wolfram Language 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, ... (OEIS 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 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, ... (OEIS 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, ... (OEIS A001274), which have common values , 2, 8, 48, 80, 96, 128, 240, 288, 480, ... (OEIS 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, ... (OEIS 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, ... (OEIS A001229).

REFERENCES:

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